LIBRARY 

OF   THE 

UNIVERSITY  OF  CALIFORNIA. 
Cla&s 


ENTKOPY; 

OK, 

rHERMODYNAMICS  FROM  AN  ENGINEER'S 
STANDPOINT. 


ENTRO  PY; 


OR. 


THERMODYNAMICS  FROM  AN  ENGINEER'S 
STANDPOINT, 


AND   THE 


REVERSIBILITY  OF  THERMODYNAMICS, 

BY 

JAMES    SWINBUKNE. 


or  THJ* 
UNIVERSITY 


NEW  YORK: 
E.    P.    DUTTON    &    CO. 


PEEFACE. 


MY  reason  for  adding  another  little  book  on 
thermodynamics  to  those  in  existence  is  that  it  is 
wanted.  As  far  as  I  am  aware  there  is  not  any 
work  on  the  steam-  or  gas-engine  in  this  country 
that  gives  a  correct  definition  of  entropy.  I  do 
not  know  about  foreign  text-books  on  this  subject. 
This  is  rather  a  serious  indictment;  and  when  it 
is  added  that  most  treatises  on  Physics,  English 
and  foreign,  contain  incorrect  definitions  of  entropy, 
and  that  even  some  of  those  specially  on  thermo- 
dynamics are  at  fault,  especially  in  classing  entropy 
as  a  factor  of  heat,  there  is  no  need  for  any  apology, 
at  least,  for  the  appearance  of  a  book  like  this. 

Instead  of  treating  the  subject  in  the  orthodox 
way,  however,  I  have  ventured  on  a  new  method 
of  explaining  it.  The  reasons  for  adopting  this 
method  are  given  in  the  paper,  "  The  Rever- 
sibility of  Thermodynamics,"  at  the  end  of  the 
book.  This  paper  was  written  to  be  criticised  by 

)    O  f\  f*  ,-» 

Ioub77 


vi  PREFACE. 

those  already  well  up  in  the  subject,  and  was  read 
before  Section  A.  of  the  British  Association  this 
year,  1903.  Though  the  paper  was  circulated  in 
proof  there  was  no  relevant  discussion.  It  gives 
a  sort  of  syllabus  of  the  method  of  exposition 
adopted  in  the  body  of  the  book,  and  the  reasons 
for  its  adoption. 

The  definition  of  heat,  which  makes  what  is 
usually  denoted  by  U  the  heat  of  the  body,  is  an 
entirely  new  departure,  as  far  as  I  know,  though 
it  is  quite  consistent  with  the  treatment  of  chemical 
energy  in  thermodynamics.  If  it  were  not  it  would 
be  wrong ;  but  its  being  consistent  with  correct 
usage  does  not  mean  that  it  is  not  new.  Another 
new  development  is  in  connection  with  the  entropy 
temperature  diagram.  It  is  assumed  in  treatises 
on  this  subject  that  the  entropy  temperature  dia- 
gram is  of  the  same  area  as  the  indicator  diagram. 
The  essence  of  the  treatment  here  followed  is  that 
it  is  not,  and  that  the  difference  is  an  indication 
of  the  badness  of  the  engine.  I  am  not  the  first, 
however,  to  call  attention  to  the  inaccuracies 
current  about  the  entropy  temperature  diagram. 
Marchis  ("  Comptes  Eendus,"  132,  p.  671)  has 
criticised  the  ordinary  error,  and  the  late  Bryan 
Donkin,  in  the  preface  to  his  translation  of 


PKEFACE.  vii 

Boulvin's  exhaustive  book,  "  The  Entropy  Dia- 
gram," says  it  "  ought  only  to  be  applied  to 
reversible  cycles";  but  he  is  not  at  all  clear. 

The  main  portion  of  this  book  appeared  as  a 
series  of  articles  in  Engineering,  August  28th , 
1903,  and  the  following  issues.  I  have  to  thank 
the  proprietors  of  Engineering  for  permission  to 
republish  the  articles. 


JAMES   SWINBUKNE. 


82,  VICTORIA  STREET, 

WESTMINSTER, 
October,  1903. 


CONTENTS. 


CHAPTER    I. 

INTRODUCTION. 

PAGE 

Origin  of  Common  Error  as  to  Entropy — Personal — 

Irreversible  Changes  not  Considered  Properly        .         1 

CHAPTER    II. 

ENTROPY. 

Work,  Heat,  and  Waste— "What  Entropy  is  Not,  and 
What  it  Is — Three  Kinds  of  Perpetual  Motion — 
Three  Laws  of  Thermodynamics  —  Reversible 
Cycle — Entropy  as  a  Factor  of  Incurred  Waste — 
Reversibility  —  Localisation  of  Entropy  —  More 
Convenient  Definition  of  the  Entropy  of  a  Body  .  6 

CHAPTER    III. 

THE  TEMPERATURE  ENTROPY  DIAGRAM. 

Irreversible  or  Free  Expansion — Difference  of  Areas 
of  6  <f>  and  p  v  Diagrams — Ungained  Work — Dia- 
gram for  Steam — Real  Factors  of  Heat  .  .  .73 


x  CONTENTS. 

CHAPTER    IV. 

CONDUCTION. 

PAGE 

Movement  of  Entropy — Conduction  of  Heat — Volume 
Growth  of  Entropy  in  Conduction— Unit  of  En- 
tropy— Physical  Meaning  of  Entropy — Conclusion  94 


APPENDIX. 

THE  REVERSIBILITY  OF  THERMODYNAMICS. 

Present  or  Orthodox  Treatment — Skeleton  of  Proposed 
Treatment  of  Thermodynamics— Waste — Reversi- 
bility —  Equilibrium  —  Stabilities  —  Conduction — 
Units — Conclusion.  113 


ENTROPY; 

OR, 

THERMODYNAMICS   FROM    AN    ENGINEER'S 
STANDPOINT, 


CHAPTER  I. 

INTRODUCTION. 

IT  may  be  as  well  to  begin  by  stating  that  I  have 
no  peculiar  theories  as  regards  entropy,  and  that  I 
am  merely  trying  to  explain,  in  as  lucid  a  way  as  I 
can,  the  underlying  principle,  using  the  term  as  it 
is  employed  in  the  thermodynamics  of  reversible 
and  irreversible  changes.  The  discussion  of  entropy 
involves  writing  a  sort  of  introduction  to  thermo- 
dynamics, so  it  may  be  worth  while  to  do  so  from 
an  engineer's  point  of  view.  The  treatment  of  the 
subject  may  thus  be  novel,  but  novelty  is  not 
important ;  the  real  aim  is  to  make  the  theory 
clear. 

Books  on  thermodynamics  do  not  usually  give 
any  idea  of  any  sort  of  physical  meaning  of  entropy, 
and  only  define  it  so  that  its  magnitude  can  be 

E.  B 


2  INTRODUCTION. 

calculated.  Many  otherwise  scientific  men,  includ- 
ing some  engineers,  have,  unfortunately,  got  hold 
of  an  inaccurate  definition  of  entropy,  derived  from 
books  which  treat  exclusively,  or  almost  exclusively, 
of  reversible  processes ;  and  as  no  properly  consti- 
tuted engineer  can  use  mathematical  symbols  with- 
out having  an  underlying  physical  idea,  a  physical 
interpretation  has  been  forced  on  to  the  inaccurate 
definition,  and  entropy  has  been  defined  as  "  heat- 
weight,"  or  as  the  quantity  factor  of  heat-energy, 
as  if  temperature  corresponded  with  head  or  differ- 
ence of  level,  and  entropy  with  weight  of  water  in 
hydraulics,  so  that  the  product  of  the  two,  in  the 
case  of  constant  level  or  temperature,  gives  energy 
or  heat.  Similarly,  the  entropy  temperature  is 
nearl}7  always  called  a  e<  heat "  diagram,  whereas  its 
area  is  not  proportional  to  any  particular  quantity 
of  existing  heat. 

In  spite  of  the  inaccuracies  of  thought  and  state- 
ment that  are,  unfortunately,  very  common,  the 
0  <£  diagram  is  of  the  utmost  value  to  steam  and 
gas  engineers,  and  my  object  is  to  try  to  get  clearer 
ideas  as  to  the  real  meaning  of  the  diagram. 
Neither  is  the  0  <#>  diagram  as  commonly  used 
seriously  wrong.  Most  writers,  after  defining 
entropy  incorrectly,  unconsciously  depart  from  their 
definitions  in  dealing  with  the  0  <j>  diagram,  so  that 
the  two  errors  approximately  cancel  out.  There  is, 
however,  still  an  error  in  the  ordinary  treatment  of 


PERSONAL.  3 

this  diagram.  But  perhaps  the  chief  evil  is  the 
confusion  of  mind  into  which  readers  are  led  by 
inaccuracy  and  inconsistency  on  the  part  of  writers 
on  the  steam-engine. 

Personal. — As  a  young  man  I  tried  to  read  ther- 
modynamics, hut  I  always  came  up  against  entropy 
as  a  brick  wall  that  stopped  my  further  progress. 
I  found  the  ordinary  mathematical  explanation,  of 
course,  but  no  sort  of  physical  idea  underlying  it. 
No  author  seemed  even  to  try  to  give  any  physical 
idea.  Having  in  those  days  great  respect  for  text- 
books, I  concluded  that  the  physical  meaning  must 
be  so  obvious  that  it  needs  no  explanation,  and  that 
I  was  especially  stupid  on  that  particular  subject. 
(Everyone  who  studies  by  himself  knows  he  is  par- 
ticularly stupid  in  certain  directions,  and  is  con- 
stantly realising  new  limitations.)  After  a  few 
3^ears  I  would  tackle  the  subject  again,  and  always 
I  was  brought  up  dead  by  the  idea  of  entropy.  I 
asked  people,  but  I  never  met  any  one  who  could 
tell  me,  and  I  met  one,  an  engineer,  who  admitted 
he  did  not  know.  Not  only  could  I  get  no  physical 
idea  of  entropy,  but  the  definition  of  entropy,  and 
the  statements  about  it,  did  not  make  sense  as 
soon  as  one  tried  to  understand  irreversible  changes. 
Later  on,  instead  of  making  the  common  mistake 
that  elementary  books  are  easy  to  understand,  I 
got  into  the  stud}7  of  irreversible  thermodynamics 
by  the  road  of  physical  chemistry,  and  found  that 

B2 


4  INTRODUCTION. 

my  previous  troubles  were  due  to  inaccurate  defini- 
tions and  faulty  analogies  on  the  part  of  writers 
who  had  an  incomplete  grasp  of  thermodynamics. 
Having  once  got  accurate  and  consistent  definitions, 
it  is  not  so  difficult  to  get  some  sort  of  physical 
idea  of  entropy.  I  hope  I  may  be  pardoned  these 
personal  reminiscences  ;  they  are  to  show  that  when 
I  write  to  correct  errors  in  scientific  text-books  and 
papers  on  engineering,  written  by  able  men,  I  do 
not  do  so  in  any  spirit  of  superiority. 

Irreversible  Changes  not  Considered. — The  source 
of  the  whole  trouble  is  the  obscurity  of  the  mathe- 
matical writer,  especially  when  he  is  not  of  the  first 
rank.  The  early  books  on  thermodynamics,  and  most 
of  the  treatises  specially  on  thermodynamics  of  the 
present  day,  are  so  anxious  to  prove  the  second  law 
with  the  help  of  the  ideal  reversible  cycle,  and  to 
explain  Kelvin's  absolute  thermometer  scale,  that 
they  discuss  reversible  changes  almost  exclusively, 
and  either  neglect  irreversible  changes,  or  treat  only 
one  case — that  of  conduction  of  heat,  and  that  in  a 
meagre  way — so  that  the  reader  gets  quite  a  wrong 
notion  of  the  subject.  Worse  than  this,  the  writer 
on  thermodynamics  makes  statements  which  are 
true  of  reversible  but  untrue  of  irreversible  changes, 
as  if  they  were  quite  general.  The  reader  who  is 
already  acquainted  with  his  subject  can  mentally 
insert  "  in  the  ideal  case  of  reversibility  only," 
every  here  and  there,  when  he  studies  thernio- 


IRREVERSIBLE    CHANGES.  5 

dynamics  ;  but  that  there  is  great  confusion  on  the 
subject,  especially  of  entropy,  is  almost  entirely  the 
fault  of  the  mathematical  writer  on  thermodynamics. 
He  makes  general  statements  which  would  only  be 
true  in  an  ideal  and  impossible  case,  and  makes  no 
attempt  to  give  any  physical  ideas  of  the  quantities 
his  symbols  represent.  To  an  address  to  the 
Institution  of  Electrical  Engineers  I  added  a  foot- 
note calling  attention  to  the  fact  that  most  engineer- 
ing and  physical,  as  opposed  to  thermodynamical, 
writers  use  a  definition  of  entropy  that  is  numerically 
accurate  in  the  hypothetical  case  of  reversibility 
only,  and  is,  in  fact,  always  wrong.  I  thought 
merely  calling  attention  to  the  error  and  its  source 
would  correct  it  at  once,  but  instead  of  that  it  led 
to  an  extraordinary  correspondence,  which  aston- 
ished me  beyond  measure  by  showing  that  many 
able  men  have  the  most  confused  notions  on  the 
subject.  I  wrote  an  article  on  Entropy  in  the 
Electrical  Review  of  January  9th,  1903,  attempting 
to  give  a  general  explanation  so  as  to  cover  chemistry 
as  well  as  physics  and  engineering ;  but  the  article 
was  written  under  pressure,  and  was  therefore  short, 
so  perhaps  it  may  do  no  harm  to  write  more  fully, 
dealing  with  the  subject  from  an  engineer's  point 
of  view,  and  discussing  the  meaning  of  the  0  </> 
diagram.  In  this  way  but  little  of  the  same  ground 
is  covered,  and  it  is  always  well  to  approach  a 
subject  from  two  different  points  of  view. 


CHAPTER   II. 

ENTROPY. 

Work,  Heat,  and  Waste. — Energy  is  indestruc- 
tible ;  but  it  exists  in  many  forms,  such  as  potential, 
kinetic,  electric,  magnetic  energy  on  one  hand,  and 
heat  on  the  other.  All  of  these,  except  heat,  are 
interchangeable,  with  a  slight  change  into  heat, 
which  can  be  diminished,  in  fact,  till  very  small, 
and  in  idea  to  nothing.  All  can  be  changed  into 
heat ;  but  heat  can  only  be  partly  changed  into  the 
other  forms.  For  want  of  a  better  word,  and  for 
simplicity,  as  "  energy  "  includes  heat,  we  m:»y  call 
the  high-grade  energy,  such  as  potential,  kinetic, 
electric,  magnetic,  &c.,  "  work,"  and  low-grade 
"heat."  We  need  not  concern  ourselves  with  such 
things  as  energy  of  radiation  or  chemical  energy 
here.  It  is  the  aim  of  the  engineer,  when  dealing 
with  steam,  gas,  oil,  or  compressing  engines,  to 
avoid  converting  or  degrading  energy  into  heat. 
The  term  "  dissipation  of  energy  "  is  generally  used 
to  denote  degradation  of  work  into  heat.  As  Ber- 
trand  points  out,  the  energy  is  not  dissipated,  as 
it  is  not  annihilated — it  is  degraded,  and  we  may 
therefore  adopt  his  word,  though  dissipated  does 


WOKK,  HEAT,  AND  WASTE.     7 

\ 

not  really  mean  annihilated.  When  work  is  con- 
verted into  heat,  it  may  be  called  "degraded"; 
but  generally  some  of  the  heat  can  be  elevated  back 
into  work  if  the  rest  is  given  out  at  a  lower  tempera- 
ture. The  portion  of  the  heat  that  must  at  least 
be  finally  given  out  as  heat  at  the  lowest  available 
temperature  is  of  no  use  :  it  is  waste.  I  will  there- 
fore call  the  part  of  the  heat  that  is  eventually  and 
unavoidably  produced  or  left  at  the  lowest  available 
temperature  the  "waste." 

The  idea  of  degradation  of  energy- — that  is,  con- 
version of  work  into  heat — is  quite  familiar  and 
clear.  Thus  all  movement  against  friction  means 
degradation.  But  there  may  be  a  change  during 
which  no  work  is  actually  converted  into  heat,  but 
which  cannot  be  unmade,  so  as  to  bring  the  "  work- 
ing substance  "  back  to  its  original  state  without 
involving  degradation.  Such  a  change  thus  "  lets 
us  in  "  for  some  degradation  at  some  future  date, 
so  that  the  degradation  is,  as  it  were,  a  liability 
incurred,  which  must  be  eventually  liquidated. 

Wliat  Entropy  is  Not,  and  What  it  Is. — In  order 
to  get  a  clear  idea  as  to  what  entropy  is,  it  is  best 
to  begin  by  stating  clearly  what  it  is  not.  It  is  not 
any  form  of  energy,  nor  a  quantity  of  the  dimen- 
sions of  energy.*  It  is  not  heat- weight.  It  is  not 

*  The  misapplication  of  the  term  "  entropy  "  by  Maxwell 
and  others,  with  no  confusion  of  idea  or  inaccuracy  of 
thought,  has  been  corrected  long  ago,  and  may  be  dismissed. 


8  ENTROPY, 

equal  to  fd  H/0,  as  in  fact  it  is  essentially  greater. 
It  is  not  heat  taken  in  by  the  substance  divided  by 
the  absolute  temperature ;  it  is  always  greater. 
It  is  not  a  factor  of  heat.  It  is  not,  in  fact,  a 
function  of  the  heat  taken  in  by  the  substance. 
The  temperature-entrop}r  diagram  is  not  a  heat 
diagram  at  all.  Its  area  does  not  represent  the 
heat  of  the  substance,  nor  the  heat  taken  in  by  the 
substance — it  is  necessarily  greater — nor  the  energy 
of  the  substance,  nor  the  energy  taken  in  by  the 
substance.  The  statements  here  contradicted,  with 
others  equally  misleading,  are  continually  made, 
not  in  advanced  treatises  on  thermodynamics,  but 
in  books  on  mathematical  physics,  and  treatises  on 
the  steam-engine.  If  anyone  who  has  not  a  clear 
idea  of  entropy  will  first  banish  all  preconceived 
notions  as  to  heat-weight,  heat  diagrams,  or  heat 
divided  by  temperature,  and  will  think  of  entropy 
as  the  measure  of  waste  actually  effected,  or  inevit- 
ably to  be  effected  subsequently,  he  will  have  a  clear 
idea.  Entropy  may  be  defined  thus :  Increase  of 
entropy  is  a  quantity  which,  when  multiplied  by  the 
lowest  available  temperature,  gives  the  incurred 
waste.  In  other  words,  the  increase  of  entropy 
multiplied  by  the  lowest  temperature  available  gives 
the  energy  that  either  has  been  alread}T  irrevocably 
degraded  into  heat  during  the  change  in  question, 
or  must,  at  least,  be  degraded  into  heat  in  bringing 
the  working  substance  back  to  the  standard  state. 


WHAT    IT   IS,   ANDT'WT.  9 


The  last  part  of  this  definition  must  not  be  omitted. 
The  entropy  of  a  body  may  increase  while  the 
entropy  of  another  body  that  gives  it  heat  decreases 
equally.  In  this  case  there  is  no  increase  of  entropy 
of  the  whole  system  and  no  new  incurred  waste. 
By  increase  of  entropy  here  is  meant  the  increase 
of  entropy  as  a  whole  in  any  isolated  system,  or 
region,  which  has  no  exchange  of  energy  with  any- 
thing outside  it  during  the  increase  of  entropy. 
When  the  increase  of  entropy  of  one  body  is  exactly 
balanced  by  the  corresponding  decrease  in  another, 
the  increase  of  entropy  of  the  body  is  called  "  com- 
pensated," and  there  is  no  real  increase  of  entropy 
and  no  further  incurred  waste  beyond  that  due  to 
the  original  production  of  the  entropy.  At  first 
sight  anyone  may  say,  "  if  the  object  is  to  measure 
the  work  irrevocably  wasted  as  heat,  why  should 
we  not  have  a  quantity  of  the  same  dimensions  as 
energy,  which  would  give  us  the  work  or  heat  right 
off,  instead  of  a  quantity  like  entropy,  which  is 
not  energy,  and  has  to  be  multiplied  by  a  tempera- 
ture to  give  the  energy  wasted?"  The  answer  is, 
that  the  practical  value  of  heat  depends  on  the 
temperature,  so  that  unless  the  lowest  available 
temperature  is  known,  the  energy  that  at  least  must 
be  run  to  waste  as  heat  is  not  known.  Suppose  by 
some  process,  say,  10,000  units  of  heat  were  pro- 
duced at,  say,  1,000°  absolute  temperature  ;  it  does 
not  in  the  least  follow  that  10,000  units  of  work  are 


10  ENTROPY. 

wasted  as  heat,  for  1,000°  may  not  be  the  lowest 
available  temperature.  Suppose  500°  is,  then,  as  will 
be  shown  directly,  half  this  heat  can  theoretically 
be  converted  back  again  into  work,  and  10  X  500 
=  5,000  units  of  work  are  really  irrevocably  degraded 
into  heat  or  wasted.  Again,  if  250°  is  the  lowest 
available  temperature,  three-quarters  of  the  10,000 
units  can,  theoretically,  be  got  back  into  work, 
10  X  250  being  wasted.  It  is  clear,  therefore,  that 
the  total  quantity  of  work  wasted  cannot  be  deter- 
mined unless  we  know  the  lowest  temperature  avail- 
able. What  we  can  know,  however,  is  the  increase 
of  entropy,  a  quantity  which,  when  multiplied  by  the 
lowest  available  temperature,  gives  the  work  wasted. 
In  this  case  we  might  call  the  entropy  10. 

Suppose  by  some  other  process  a  change  took 
place  without  the  actual  degradation  of  work  into 
heat,  but  of  such  a  nature  that  to  get  the  substance, 
for  instance,  some  wire-drawn  steam,  back  into  its 
original  state,  10,000  British  thermal  units  would 
eventually  have  to  be  converted  into  heat ;  if  this 
heat  is  given  out  into  a  body  at  a  temperature  of 
1,000°,  and  500°  is  available  as  a  lowest  tempera- 
ture, half  can  again  be  theoretically  saved,  and 
so  on.  In  this  case,  however,  the  increase  of 
entropy  takes  place  when  the  steam  is  wire-drawn ; 
its  growth  is  not  delayed  until  the  heat  is  actually 
produced. 

As  will  be  more  fully  explained  directly,  we  can 


PERPETUAL   MOTION.  11 

only  discuss  increase  of  entropy,  or  decrease  of 
entropy  of  a  body.  We  cannot  evaluate  the 
whole  entropy  of  a  bod}'.  The  entropy  of  a 
body  is  therefore  measured  by  comparison  with  a 
standard  state.  For  instance,  if  water  at  32°  Fahr. 
is  taken  as  a  standard  or  zero  for  water  and 
steam,  the  entropy  of  steam  is  really  the  difference 
between  its  entropy  and  that  of  the  same  weight  of 
water  at  32°  Fahr. 

Three  Kinds  of  Perpetual  Motion.  —  We  can 
certainly  take  it  as  a  fundamental  principle  that 
perpetual  motion  is  impossible.  The  original  idea 
of  perpetual  motion  was  a  mechanism  that  never 
stopped.  It  was  not  necessarily  a  mechanism  that 
created  energy,  or  gave  out  energy.  We  may  also 
take  it  that  energy  is  conserved,  or  invariable 
in  quantity.  The  idea  of  perpetual  motion  that 
involves  creation  of  energy  thus  contradicts  the 
law  of  conservation  of  energy;  but  a  frictionless 
mechanism,  which  would  run  for  ever,  is  merely 
unrealisable.  It  is  not  an  absurdit}r :  it  is  a  theo- 
retical abstraction.  Every  mechanism  has  friction. 
But  imagine  a  mechanism  in  a  case  through  which 
no  energy  passes,  so  that  the  energy  inside  the  case 
is  constant.  Once  started,  the  friction  would  con- 
vert work  into  heat ;  but  if  the  mechanism  could 
convert  the  heat  so  produced  completely  into  work 
again,  there  would  be  no  contradiction  of  the  law 
of  conservation  of  energy,  as  the  energy  is  constant. 


12  ENTROPY. 

There  would  be  no  stoppage  by  friction,  as  all  tbe 
work  converted  into  heat  would  be  converted  into 
work  again,  as  kinetic  energy  of  motion.  We 
would  thus  have  a  form  of  perpetual  motion,  not 
involving  frictionless  mechanism,  and  not  involving 
creation  of  energy.  Does  this  form  of  perpetual 
motion  rank  with  the  creation  of  energy  form,  in 
which  the  mechanism  gives  out  work  and  creates 
energy;  or  is  it  a  theoretical  abstraction,  which 
cannot  be  realised  only  because  of  the  imperfection 
of  our  workmanship,  like  the  frictionless  mechanism  ? 
The  answer  is  that  it  is  not  a  mere  theoretical 
abstraction,  but  an  absurdity  of  the  same  order 
as  the  energy-creating  form  of  perpetual  motion. 
It  has  been  called  "  perpetual  motion  of  the  second 
class."  It  is  perpetual  motion  which  does  not 
involve  creation  of  energy,  but  involves  the  opposite 
or  complete  undoing  of  the  degradation  of  energy. 
The  friction  degrades  work  into  heat.  Some  of  the 
heat  might  be  converted  back  into  work  if  the  rest 
is  given  out  at  a  still  lower  temperature ;  but  to 
make  this  mechanism  work  continuously  in  spite  of 
its  friction,  all  the  heat  would  have  to  be  converted 
back  into  work. 

The  waste  may  be  defined  as  the  residual  heat, 
of  which  none  can  be  elevated  back  into  work ;  that 
is  to  say,  it  is  the  heat  that  must  still  remain  at  the 
lowest  available  temperature.  Perpetual  motion  of 
a  mechanism  with  friction  would  involve  the  whole 


PERPETUAL   MOTION.  13 

of  the  heat  produced  by  friction  being  elevated  back 
into  work.  It  would  thus  involve  the  reduction  or 
diminution  of  waste  once  incurred.  This  abundant 
experience  shows  it  to  be  an  absurdity  of  the  same 
order  as  the  creation  of  energy.  We  may  then 
say: — 

Energy  is  Conservative  or  constant  in  quantity  in, 
the  universe,  or  in  any  part  of  it  which  neither 
takes  in  or  gives  out  energy.  This  is  the  first  law 
of  thermodynamics,  as  it  involves  the  equivalence 
of  heat  and  work. 

Waste,  once  incurred,  cannot  be  diminished  in  the 
universe,  or  in  any  part  of  it,  which  neither  takes 
in  nor  gives  out  energy.  This  is  the  second  law  of 
thermodynamics. 

The  relations  of  perpetual  motion  to  thermo- 
dynamics may  be  summed  up  shortly. 

1.  Perpetual  motion  where  an  otherwise  isolated 
system  gives  out  energy  continually  is  impossible, 
and  not  even  approximately  realisable.     This,  with 
the  proviso  that  an  otherwise  isolated  system  cannot 
absorb  energy  without  increasing  its  internal  energy, 
is  the  principle  of  the  conservation  of  energy,  or 
first  law  of  thermodynamics. 

2.  Perpetual  motion  of  an  isolated  system,  such 
as  a  mechanism  with  friction,  is  impossible,  and  not 
approximately  realisable.     This  is  the  second  law 
of  thermodynamics. 

3.  Perpetual  motion  of  a  frictionless  mechanism 


14  ENTROPY. 

is  unrealisable.  This  may  be  called  the  third  law 
of  thermodynamics. 

As  every  change  in  existence  either  involves 
some  friction  or  some  analogous  cause  of  waste, 
every  change  or  process  causes  some  increase  of 
waste.  A  change  or  process  which  has  no  friction, 
and  nothing  analogous  to  friction,  is  an  ideal 
abstraction  only,  not  an  absurdity,  like  the  creation 
of  energy,  but  a  useful  hypothetical  case  which  can 
be  approximately  realised.  A  frictionless  mechanism 
is  the  simplest  example. 

The  notion  of  degradation  and  of  waste  is  here 
introduced,  not  as  being  convenient  for  practical 
use,  but  as  a  good  way  of  getting  at  the  significance 
of  entrop3T.  The  difficulty  with  waste  is  that  you 
cannot  give  it  a  value  unless  you  know  the  lowest 
available  temperature. 

Reversible  Cycle. — It  is  shown  in  text-books  that 
if  an  imaginary  thermodynamic  engine,  with  a  per- 
fect gas  as  working  substance,  working  a  Carnot 
cycle  without  friction  or  other  cause  of  waste,  takes 
in  heat  HI  from  a  reservoir  or  body  so  large  that  it 
can  give  up  this  heat  without  fall  of  temperature,  at 
temperature  Ol  measured  with  a  perfect  gas  thermo- 
meter, and  gives  out  heat  H2  to  another  reservoir 
at  02,  the  engine  returning  to  its  original  condition, 
so  that  it  has  performed  a  cycle,  a  certain  proportion 
of  the  heat  HI  will  be  elevated  into  work,  and  the 
rest  must  be  rejected  as  H2  at  the  lower  temperature 


REVERSIBLE    CYCLE.  15 

the  engine  has,  by  hypothesis,  returned  to  its 

i  TT  f\ 

\  state.      It  is  also  shown  that  — -  =  -1  in 


particular  case. 

It  must  be  noticed  that  H  here  is  not  the  in- 
1  crease  of  heat  of  the  gas;  it  is  the  decrease  of 
heat  of  the  reservoir ;  or,  to  put  it  more  generally, 
it  is  the  heat  that  comes  into  the  gas  from  the 
outside.  This  heat  may  be  converted  into  work 
at  once,  so  that  the  heat  of  the  gas  is  not  in- 
creased. Again  the  heat  of  the  gas  may  be  varied 
without  any  heat  passing  in  or  out  through  its  case 
or  envelope  ;  for  instance,  by  adiabatic  compression 
or  expansion. 

0  is  here  the  temperature  of  the  envelope  the 
heat  H  passes  through.  In  reversible  changes  like 
this  the  reservoir  and  gas  are  at  the  same  tempera- 
ture, and  so  is  the  envelope,  so  0  is  the  temperature 
of  all  three.  But  if  the  temperature  of  the  working 
substance  and  the  body  that  supplies  heat  H  are 
not  uniform,  0  denotes  the  temperature  of  the  sur- 
face of  the  working  substance  through  which  the 
heat  passes.  This  is  important,  and  laxity  of  state- 
ment in  books  on  thermodynamics  has  given  rise  to 
much  confusion  in  this  connection.  When  0  is 
mentioned  hereafter  as  the  temperature  of  the 
body,  the  body  and  envelope  are  at  the  same 
temperature. 

The  machine  in  the  ideal  cycle  with  perfect  gas, 


16 


ENTROPY. 


starting  from  A,  in  the  p  v  diagram,  Fig.  1,  where 
the  gas  is  at  temperature  0.  and  pressure  and 
volume  pi  Vi  takes  in  heat  HI,  while  it  slowly 
expands  still  at  temperature  6\.  When  it  has 
expanded  to  B,  it  has  thus  taken  in  heat  HI,  and 
converted  it  all  into  external  work.  The  gas  has 
expanded  at  constant  temperature,  and  Joule's 
experiment  showed  that  a  real  gas  needs  practically 
no  heat  to  expand  it ;  a  perfect  gas  is  supposed  to 
absorb  no  heat  in  expansion.  From  B  to  C  the 


o  "™" 


f       G  H 


.gas  expands  without  taking  in  any  heat,  and  as  it 
does  external  work  its  temperature  falls  to  02.  From 
C  to  D  it  is  compressed,  giving  out  heat  H2  at 
temperature  02,  and  D  is  chosen  so  that  from  it  the 
gas  is  compressed  without  giving  out  any  heat  until 
it  arrives  at  A.  As  the  gas  has  the  same  internal 
energy,  Ui,  at  A  and  B,  and  the  same,  U2,  at  C  and 
D,  as  it  merely  changes  volume,  but  not  tempera- 
ture, from  A  to  B,  and  C  to  D,  the  difference 
between  the  internal  energy  of  A  B  and  C  D  is  the 
same,  whether  the  change  is  by  the  path  B  C  or 


REVERSIBLE    CYCLE. 


17 


AD.  The  work  done  by  the  gas  from  B  to  C  is 
therefore  equal  to  that  from  A  to  D.  Suppose, 
therefore,  that  when  the  gas  has  reached  B  at  tem- 
perature 0i,  instead  of  cooling  it  by  letting  it  give 
out  its  energy  as  work,  we  cool  it  by  abstracting 
heat,  without  expanding  it ;  it  will  then  fall  to  E 
(Fig.  2),  as  the  internal  heat  energy  of  a  perfect  gas 
varies  directly  as  the  temperature.  (To  make  this 


Jf 


quite  accurate  I  have  assumed  a  perfect  gas  thermo- 
meter, whose  readings  differ  very  slightly  from  a 
mercury  instrument.)  The  height  of  E  will  be  to 
that  of  B  as  0%  is  to  0i.  Now  compress  the  gas  to 
F,  giving  out  heat  H2 ;  then  give  back  exactly  the 
heat  that  was  abstracted  from  B  to  E,  and  the 
temperature  rises  to  6\  at  A  again.  It  is  clear  that 
the  height  of  F  is  to  that  of  A  also  as  02  is  to  0i, 
and  that  is  true  of  any  pressure  lines  we  may  draw. 
Any  vertical  slice  I  J  K  L  of  the  area  A  B  G  H  is 
E.  c 


18  ENTROPY. 

therefore  cut  by  the  curve  F  E,  so  that  the  relation 
of  the  whole  to  the  lower  part  is  #i/02,  and  the  ratio 
of  the  area  A  B  G  H  or  HI  to  F  E  G  H,  or  H2  is 
as  01/02,  and  that  of  A  B  E  F  the  balance  of  external 
work  Wz,  to  A  B  G  H  or  HI  is  (^  -  02)/0i,  Garnet's 
ratio.  As  A  B  G  H  is  equal  to  the  work  done  on 
expansion,  and  therefore  to  HI,  and  E  F  H  G  to 
the  work  put  in  on  compression,  given  out  as  H2, 
A  B  E  F  is  equal  to  the  balance  of  external  work 
done,  or  to  the  heat,  HI  —  H2  converted  into  work. 
The  output  of  the  engine  cycle  is  thus,  where  W  b  is 
the  balance  of  work  done  :— 


=  H!  -  Ha  = 

and  the  efficiency 


W*  _  Ol  -  02 
Hi"       #i 

and 

Hi_0i 
H2  ~~  02' 

The  proof  is  not  conclusive  as  it  stands,  because 
though  the  heat  given  out  by  the  gas  from  B  to  E 
is  the  same  as  that  taken  in  from  F  to  A,  it  might 
be  said  that  the  temperatures  vary,  so  that  the  heat 
might  not  be  taken  in  and  given  out  at  the  same 
temperature.  The  lines  E  B,  F  A  can  be  divided 
into  a  number  of  equal  corresponding  parts,  each 
part  of  E  B  having  a  corresponding  part  of  F  A. 
Each  pair  of  parts  then  represents  a  corresponding 
step  in  temperature  due  to  the  same  increase  or 


REVERSIBLE    CYCLE. 


19 


decrease  of  heat.  A 
perfect  gas  thermometer 
may  be  made  which  reads 
in  pressure  of  gas  at 
constant  volume.  If  the 
gas  in  the  cylinder  is  so 
used,  the  lines  F  A  and 
G  B  would  be  divided 
up  into  0  equal  parts, 
and  F  A  would  have 
0i  —  0%  equal  parts,  and 
so  would  E  B.  The 
heat  capacity  of  a  perfect 
gas  being  constant,  each 
of  these  degrees  means 
the  same  heat  taken  in 
at  the  same  temperature. 
The  proposition  can  be 
proved  otherwise.  Fig. 
3  shows  a  double  ther- 
modynamic  engine,  with 
two  cylinders  which  may 
be  arranged  back  to 
back.  The  working  sub- 
stance is  in  the  cylinder 
to  the  right.  First  the 
gas  G  expands,  without 

using  the  left-hand  engine,  at  constant  temperature, 
the   piston   moving   from   the    dotted   position   A, 

c2 


20  ENTROPY. 

to  B.  Then  the  heat  reservoir  E  is  removed  from 
the  cover  of  the  cylinder,  and  the  second  cylinder  S 
applied.  The  gas  in  S  is  then  slowly  expanded, 
taking  heat  from  the  working  substance,  and  also 
supplying  heat  of  its  own.  All  this  heat  is 
converted  into  work.  When  the  substance  is  thus 
cooled  to  0%,  the  second  cylinder  is  removed,  and  a 
cold  reservoir  'applied.  The  piston  is  pushed  in 
from  B  to  A,  giving  out  heat  H2.  The  other 
cylinder  is  then  applied,  and  the  gas  G  heated  to 
its  original  state  by  compressing  the  gas  in  S  to  its 
original  state.  The  double  engine  has  then  taken 
in  heat  HI,  and  given  out  H2  and  work  Wz>,  and  there 
is  no  question  about  the  two  intermediate  steps. 
The  artist  has  designed  these  engines  without 
guides  or  brasses,  because  hypothesis  prevents  there 
being  any  friction  anyhow. 

We  have  thus  one  specimen  ideal  machine  which 
will  convert   heat   from    a   body  at   #1  into  work, 

rejecting  at  a  lower  temperature  02with  an  efficiency 

f\       c\ 
of    ]   ..    -  :   it  is  now  to  be  shown  that  this  one 

*i 
specimen    will    give     us    information    as     to    the 

possibilities  of  all  such  machines.  The  machine 
will  obviously  work  equally  well  in  either  direction. 
If  it  is  reversed,  it  takes  in  heat  H2  at  the  lower 
temperature,  converting  it  all  into  work,  and  takes 
in  more  work  at  0\  giving  out  HI.  Suppose  there 
were  any  other  possible  kind  of  thermodynamic 


REVERSIBLE    CYCLE.  21 

engine  which  could  work  with  a  higher  efficiency 

0    -  f) 
than    l  -     3,  and  suppose  it  took  in  HI  and  gave 

out  more  balance  of  work  than  Wj,  say  W,  and 
therefore  rejected  less  heat  than  H2,  say  H3;  so 
that  HI  —  H3  =  W.  Imagine  these  machines  coupled 
together.  The  new  machine  would  drive  the  old 
one  round  backwards,  as  W  >  W&,  and  at  every  turn 
the  hotter  reservoir  would  give  as  much  heat  to  the 
new  engine  as  the  old  engine  gave  it.  But  the 
cooler  reservoir  would  only  get  H3  from  the  new 
engine  every  turn,  while  the  old  engine  would  take 
away  H2.  The  difference  between  W  and  Wj  might 
then  be  devoted  to  overcoming  friction,  being  con- 
verted into  heat,  and  given  to  the  lower  reservoir. 
We  would  then  have  a  case  of  second-class  perpetual 
motion — namely,  a  mechanism,  with  friction,  which 
could  be  shut  up  in  a  case,  and  go  on  for  ever. 
The  second  law  of  thermodynamics,  which,  in  the 
form  I  have  put  it,  is  a  denial  of  the  possibility  of 
perpetual  motion  of  a  machine  which  has  any 
friction,  thus  shows  that  the  maximum  ideal  or 

r\       _    s\ 

theoretical  efficiency  of  a  heat  engine  is  -^—~ — -. 

vi 
In  the  example  given  of  a  thermodynamic  machine 

with  a   higher  efficiency  working   the    machine  of 

f\  A 

efficiency    backwards,    we     supposed    the 
*i 
balance    of    work   per   cycle    to   go    into   friction. 

Suppose,  however,  the  friction  were  reduced  a  little  ; 


22  ENTROPY, 

the  pair  of  engines  could  take  all  the  heat  from  the 
cooler  and  put  it  into  the  hotter  reservoir. 

Entropy  as  a  Factor  of  Incurred  Waste.  —  Suppose 
that  we  had  produced  the  heat  Ui*  by  degrading 
work  into  heat,  say  by  friction,  the  heat  being 
produced  in  a  body  at  temperature  Blt  and  we  want 
to  find  the  waste  incurred.  If  0%  is  the  lowest 
available  temperature,  and  as  the  thermodynamic 
engine  is  frictionless  in  its  widest  sense,  so  that 
there  is  no  increase  of  waste  incurred  by  its  working, 
the  incurred  waste  is,  when  the  heat  Ui  is  received 
by  the  engine  as  HI. 


So  that  altering  the  lowest  available  temperature 
alters  the  final  waste  in  proportion.  So  the  incurred 
waste  corresponding  to  the  wasteful  conversion  of  a 
given  quantity  of  work  into  an  equivalent  quantity 
of  heat  at  any  given  temperature  varies  inversely  as 
the  temperature  at  which  the  heat  is  formed,  and 
directly  as  the  lowest  -available  temperature.  But 

*  The  symbol  U  is  generally  used  to  denote  more  than 
mere  sensible  heat,  but  it  includes  sensible  hi-iit,  and  it  is 
used  to  denote  the  increment  of  heat  of  the  body,  while  II 
denotes  the  heat  that  passes  through  the  envelope.  On 
page  122  it  is  urged  that  U  should  be  regarded  as  the  heat  of 
the  substance.  This  involves  a  new  definition  of  heat,  which, 
though  consistent  with  orthodox'  thermodynamics,  is  itself 
unorthodox.  It  is,  therefore,  not  introduced  here. 


A  FACTOR  OF  INCURRED  WASTE.    2B 

the  entropy  has  been  defined  as  a  quantity  whose 
increase,  when  multiplied  by  the  lowest  available 
temperature,  gives  the  inevitable  incurred  waste. 
Therefore,  still  assuming  the  frictionless  thermo- 
dynamic  engine  to  be  used,  the  entropy  is  inde- 
pendent of  the  lowest  available  temperature — 

as  02<I>  =  H2 

by  definition, 

and  TT        TT  TJ 

H2  _  HI  HI 

~Z~~    'a   '  IP* 

t/2  "l  "1 

where  H  is  the  heat  taken  in,  and  0  the  temperature 
of  the  envelope  through  which  it  passes — in  this 
case  the  same  as  that  of  the  gas  and  reservoir.  If, 
therefore,  we  know  the  amount  of  heat  Ui  or  HI 
produced,  and  the  temperature  0\,  we  can  get  3>, 
and  we  need  not  trouble  about  the  ultimate  actual 
waste.  Similarly,  if  a  change  has  taken  place  which 
has,  in  fact,  produced  no  heat,  but  is  of  such  a 
nature  that  to  get  the  working  substance,  whatever 
it  may  be,  back  to  its  original  state,  work  must  be 
degraded  into  heat,  we  have  no  information  as  to 
the  waste  so  caused.  But  if  we  know  the  highest 
temperature  Q\  at  which  the  inevitable  heat  can  be 
given  out,  we  get  Hi/0i  =  $,  the  entropy ;  and  that 
gives  us  the  incurred  waste  corresponding  to  any 
lower  temperature  that  may  eventually  be  available. 
The  incurred  waste  thus  varies  directly  as  the 
lowest  available  temperature,  and  directly  as  the 


24  ENTROPY, 

entropy ;  and  if  the  lowest  temperature  is  fixed,  the 
waste  incurred  depends  simply  on  the  entropy.  If, 
therefore,  a  change  of  any  sort  involved  no  increase 
of  entropy,  it  would  involve  no  increase  of  waste. 
Waste  can  be  diminished  by  obtaining  a  lower 
available  temperature,  but  not  by  any  other  means. 
If  it  were  possible,  we  could  then  have  perpetual 
motion  of  the  second  class,  the  degradation  of  work 
by  friction  being  wholly  undone  by  the  mechanism 
itself.  This  leads  to  a  most  important  law ;  as  in 
all  real  changes  there  is  either  friction,  or  some 
analogous  source  of  waste,  every  change  whatever 
involves  waste,  small  though  it  may  be ;  and  as  the 
change  does  not  affect  the  lowest  available  tempera- 
ture, there  is  always  an  increase  of  entropy.  The 
entropy  of  the  working  substance  may  be  increased 
or  diminished  or  remain  constant,  but  the  entropy 
of  the  substance  and  its  externals  increases.  The 
entropy  of  an  isolated  system — that  is  to  say,  a 
system  which  as  a  whole  neither  takes  in  nor  gives 
out  any  energy — increases  with  every  change. 
Entropy  once  created  is  therefore  permanent  for  all 
time.  The  entropy  of  the  universe  tends  towards  a 
maximum,  and  is  always  growing  towards  it. 
Changes  take  place  in  Nature  in  which  heat  is 
produced,  as  by  friction.  Changes  take  place  in 
which  heat  disappears,  or  is  elevated  into  work,  as 
when  a  gas  expands  adiabatically  doing  external 
work,  or  when  a  photographer  dissolves  what  he  calls 


A  FACTOR  OF  INCURRED  WASTE.  25 

hypo  in  water  and  gets  a  mild  freezing  mixture.  In 
this  case  the  heat  is  not  converted  into  external 
work,  but  into  another  form  of  heat.  But  in  Nature 
no  change  takes  place  which  does  not  either  produce 
heat,  and  thus  give  rise  to  waste,  or  incur  waste, 
without  actually  producing  the  heat  there  and  then. 
Clausius's  general  statement  that  the  energy  of  the 
universe  remains  constant,  and  the  entropy  strives 
towards  a  maximum,  may  therefore  be  paraphrased 
thus: — "The  energy  of  the  universe  is  constant, 
but  no  change  takes  place  without  incurring  waste 
of  energy." 

It  is  the  business  of  the  engineer  to  design  his 
machinery  to  utilise  energy  to  the  utmost ;  and  to 
get  as  little  as  possible  into  the  waste  form  of  heat 
at  low  temperatures.  If  he  takes  care  of  his  entropy, 
the  waste  will  take  care  of  itself.  He  should  design 
his  steam- or  gas-engine  so  that  the  originarentropy 
of  the  fuel  and  oxygen  is  increased  as  little  as 
possible.  He  is  not  responsible  for  the  original 
entropy,  but  he  is  for  all  increase  of  it ;  and  the 
more  he  keeps  down  the  increase  of  entropy,  and 
the  corresponding  incurred  waste,  the  more  near 
perfection  is  his  engine. 

The  theory  of  thermodynamics  is  therefore  vitally 
important,  and  the  use  of  the  0  3>  diagram  should  be 
to  the  engineer  what  the  balance-sheet  is  to  the 
financier.  The  difference  is  that  the  financier  can 
make  either  profit  or  loss,  but  the  engineer  can 


26  ENTROPY. 

only,  and  must,  make  loss :  it  is  his  business  to 
keep  it  as  small  as  possible. 

Reversibility. — We  may  now  consider  the  question 
of  reversible  and  irreversible  changes  and  reversible 
and  irreversible  cycles. 

If  the  working  substance  were  to  change  from  the 
state  A  to  the  state  B  in  such  a  way  that  the  change 
could  be  reversed,  so  that  the  substance  went  from 
B  to  A  in  such  a  way  that  every  part  of  the  change 
from  B  to  A  were  exactly  the  same  as  the  corre- 
sponding change  from  A  to  B,  but  in  the  opposite 
direction,  the  change  or  process  would  be  reversible. 
There  are  no  reversible  changes  in  Nature ;  a  rever- 
sible change  is  an  ideal,  or  an  abstraction,  which 
can  only  be  approximately  reached.  There  is  no 
such  thing  as  a  circle  or  a  square  or  a  straight  line. 
They  are  mathematical  abstractions  which  can  be 
approached,  and  are  exceedingly  convenient  for 
reasoning  purposes.  In  a  reversible  change,  not 
only  must  the  substance  retrace  exactly  the  same 
path,  but  everything  that  is  involved  in  the  change 
must  retrace  the  same  path.  The  perfect  gas  in 
the  thermodynamic  engine  is  the  simplest  case 
of  reversibilit}' ;  each  of  the  changes  A,  B  ;  B,  C ; 
C,  D  ;  and  D,  A  being  reversible.  Heat  was  taken 
in  and  work  given  out  from  A  to  B.  From  B  to 
A,  therefore,  heat  is  given  out  and  work  taken  in, 
in  exactly  the  reverse  way. 

When   the   working   substance   starts  from   any 


REVERSIBILITY.  27 

state,  and,  after  change,  returns  to  the  original 
state,  it  is  said  to  have  performed  a  cycle.  If 
every  part  of  the  cycle  is  made  up  of  changes 
of  which  each  is  reversible,  the  whole  cycle  is 
reversible. 

A  reversible  cycle  can  cause  no  incurred  waste ; 
for  if  it  did,  it  would  reduce  incurred  waste  if 
worked  the  other  way,  and  that  would  give  us 
second-class  perpetual  motion.  Thus  a  thermo- 
dynamic  engine  working  a  reversible  cycle  might 
convert  a  certain  quantity  of  heat  HI,  taken  from  a 
reservoir  at  Oi  into  work  W,  rejecting  H2  at  a  lower 
temperature.  If  it  were  coupled  to  the  wasteful 
reversible  engine  of  a  size  to  give  out  HI  at  Oi,  each 
revolution,  that  engine,  when  working  backwards, 
and  reducing  incurred  waste,  must  take  in  more 
than  H2  at  the  lower  temperature  to  give  out  HI  at 
the  highest ;  so  it  needs  less  work  each  cycle  to 
drive  it  as  H2  +  W  =  HI.  The  balance  of  energy 
per  cycle  might  then  be  devoted  to  overcoming  the 
friction,  unavoidable  in  a  real  engine,  and  any 
balance  might  do  other  work,  and  we  would  have 
second-class  perpetual  motion. 

A  reversible  change  is,  therefore,  a  change  in 
which  there  is  no  waste  incurred,  and  in  a  reversible 
cycle  there  is  no  waste  incurred. 

But  we  have  denned  entropy  as  a  quantity  whose 
increase,  when  multiplied  by  the  lowest  available 
temperature,  gives  the  incurred  waste.  There  is 


28  ENTROPY. 

thus,  then,  no  increase  of  entropy  during  a  rever- 
sible change  or  a  reversible  cycle. 

The  entropy  of  a   working   substance    may,    of 
course,  increase    or  diminish    during   a   reversible 
change,  but  the  entropy  of    an  isolated  system— 
that  is  to  say,  the  working  substance — and  whatever 
is  affected  by  the  change,  remains  constant. 

Localisation  of  Entropy. — There  has  already  been 
incurred  an  enormous  amount  of  waste  in  the 
universe.  From  the  beginning  of  things  there  has 
been  degradation  of  work  into  heat  which  can  be 
elevated  only  partially  into  work  by  rejecting  some 
of  it  at  a  lower  temperature.  There  is  thus  an 
enormous  total  entropy  in  the  universe,  and  it  is 
always  increasing.  More  than  that,  as  work  gets 
degraded  into  heat  the  lowest  available  temperature 
is  raised,  so  that  the  incurred  waste  is  increasing, 
not  only  in  proportion  to  the  growth  of  entropy, 
but  also  in  proportion  to  the  lowest  available 
temperature.  But  how  should  entropy  be  localised  ? 
When  heat  is  produced,  say,  by  friction,  it  means 
that  something  gets  hotter,  or  else  the  heat  is 
absorbed  and  rendered  latent.  To  get  the  hotter 
substance  back  to  its  original  state  it  must  either 
give  out  the  heat  as  heat,  or  it  may  expand,  doing 
external  work,  and  cooling  to  its  old  temperature, 
being  then  compressed  to  its  original  volume  and 
rejecting  heat,  which  is  degraded  energy.  The 
increased  waste  is  clearly  connected  with  the  hot 


LOCALISATION   OF   ENTROPY.        29 

body.  The  hot  body  may  be  moved  about,  or  taken 
into  different  surroundings  ;  but  to  get  it  back  to 
its  original  state  involves  its  giving  out  waste  heat. 
We  therefore  talk  of  the  entropy  of  the  substance. 
If  the  substance  is  homogeneous  in  every  way,  it 
may  be  divided  up  into  parts,  say  seven,  and  each 
part  will  have  to  give  out  one-seventh  of  the  waste 
heat  that  the  whole  has  to  give  out  in  returning  to 
its  original  state.  Entropy  is  thus  additive,  like 
weight,  or  mass,  or  volume — not  qualitative,  like 
temperature,  hardness,  or  pressure  ;  two  tons  of 
water  at  a  given  temperature  has  twice  the  entropy 
of  one  ton,  and  so  on.  We  thus  talk  of  the  entropy 
of  a  substance — for  instance,  the  entropy  of  17  lb. 
of  steam  under  given  pressure,  and  volume,  and 
temperature.  We  can  also  talk  of  the  entropy  per 
unit  mass,  or  specific  entropy  of  a  body.  For 
instance,  in  working  0  <j>  diagrams,  the  diagram  is 
generally  drawn  for  a  pound  of  steam,  and  then 
shows  the  specific  entropy.  The  term  "  specific 
entropy  "  is  not  often  used,  but  it  is  as  well  to  adopt 
it,  to  distinguish  between  the  entropy  of  a  mass  of 
steam  and  its  entropy  per  pound.  The  example  of 
Professor  Planck  may  well  be  followed.  He  uses 
capitals  for  volume,  entropy,  and  so  on,  and  small 
letters  for  specific  volume,  and  entropy. 

It  has  been  shown  that  during  any  reversible 
change  there  is  no  incurred  \vaste,  and  therefore  no 
increase  of  entropy  in  the  system  concerned.  Thus, 


30  ENTROPY, 

if  an  isolated  system  consists  of  some  heat  reser- 
voirs, a  perfect  gas-engine,  and  a  flywheel ;  if  there 
is  a  reversible  change,  there  is  no  increase  of 
entropy  of  the  system.  The  perfect  gas  may  have 
its  entropy  increased,  however,  but  in  that  case  the 
entropy  of  some  of  the  reservoirs  is  decreased  by 
exactly  the  same  amount. 

More  Convenient  Definition  of  the  Entropy  of  a 
Body. — So  far  we  have  nearly  always  discussed  the 
entropy,  or  the  increase  of  entropy,  of  a  whole 
isolated  system.  In  thermodynamics  this  course  is 
inconvenient,  and  it  is  much  more  useful  to  discuss 
the  entropy  of  the  working  substance  alone.  When 
the  entropy  of  the  working  substance  is  increased 
by  exactly  the  amount  the  entropy  of  the  rest 
of  the  isolated  system  is  diminished,  the  increase 
is  called  "compensated."  If  the  working  substance 
increases  its  entropy  by  any  reversible  change,  then 
its  increase  of  entropy  is  compensated  by  exactly 
equal  decrease  of  entropy  in  other  things  which  are 
involved  in  the  change.  Thus  if  there  were  two 
bodies,  say  vessels  of  partly  melted  lead,  A  and  B, 
and  A  were  able,  through  an  infinitesimal  difference 
of  temperature,  to  communicate  one  thermal  unit 
to  B,  in  an  infinite  time,  the  change  would  be 
sensibly  reversible,  so  the  entropy  of  A  and  B 
together  would  be  unchanged. 

But  if  we  now  proceed  to  reduce  B  to  the  tempera- 
ture of  ice,  so  as  to  compare  it  with  the  standard 


OF   A  BODY.  31 

state,  the  first  thing  it  does  is  to  allow  the  extra 
lead  thawed  by  the  unit  of  heat  to  solidify,  giving 
out  a  thermal  unit  at  the  temperature  of  melting 
lead,  and  the  waste  corresponding  to  this  unit  of 
heat,  divided  by  the  lowest  available  temperature, 
was  the  increase  of  entropy  of  B.  It  has  already 
been  explained  that  this  is  equal  to  the  heat  taken 
in  at  the  melting-point  of  lead,  divided  at  the 
absolute  temperature  at  which  it  is  taken  in — 
namely,  1/0  of  a  unit  of  entropy,  where  0  is  the 
melting-point  of  lead.  The  entropy  is  thus  localised. 
We  may  now  give  another  definition  of  entropy, 
which  isjnpre  convenient  than  that  in  terms  of  the 
incurred  waste,  because  it  refers  to  the  entropy  of 
the  working  substance,  without  taking  the  whole 
isolated  system  into  account,  and  because  it  is  not 
given  in  terms  of  the  lowest  available  temperature 
or  the  incurred  waste.  Thejiew__definitioiLjnay  be 
arrived  at  by  considering  reversible  changes.  If  two 
bodies  A  and  B  are  in  contact,  and  are  at  the  same 
temperature,  and  there  is  a  reversible  change 
so  that  the  sum  of  the  entropies  of  A  and  B 
remains  constant,  B  can  only  increase  its  entropy 
by  a  corresponding  decrease  of  the  entropy  of  A. 
But  the  only  way  A  can  have  its  entropy  reduced  is 
by  losing  heat.  For  of  the  heat  in  A  a  part  is  avail- 
able, and  the  rest,  depending  on  the  lowest  available 
temperature,  is  waste.  This  waste  can  only  be 
reduced  by  abstracting  some  of  the  heat  of  A.  A 


32  ENTROPY, 

might  lose  work,  but  that  would  not  reduce  its 
entropy.  We  thus  find  that  the  entropy  of  a  body 
can  only  be  reduced  by  its  losing  heat.  But  it  does 
not  follow  that  if  a  body's  heat  is  decreased  it  loses 
entropy.  For  if  a  gas  is  expanded  in  a  cylinder  so 
as  to  do  external  work  without  being  supplied  with 
heat  from  the  outside,  its  heat  is  decreased.  Its 
entropy  is  not,  however,  reduced,  for,  as  already 
explained,  the  waste,  once  incurred,  cannot  be 
reduced ;  so  that  the  entropy  of  a  body  cannot  be 
reduced  without  at  least  an  equivalent  increase  ol 
the  entropy  of  something  else.  The  entropy  of  A 
can  thus  only  be  reduced  by  its  losing  heat  to  some 
other  body.  Merely  diminishing  its  heat  by  con- 
verting it  into  work,  or  giving  out  the  work,  does 
not  decrease  its  entropy.  The  heat  must  be  given 
out  so  as  to  cause  at  least  an  equal  simultaneous 
increase  of  entropy  elsewhere.  This  practically 
means  that  for  the  entropy  of  A  to  decrease,  heat 
must  pass  out  across  the  boundary  or  envelope 
of  A.* 

*  The  only  exception  that  comes  to  mind  is  the  case  where 
a  thermo-circuit  is  arranged  with  a  junction  inside  A.  li 
the  other  junction  is  outside  A,  say  in  B,  and  B  is  colder, 
heat  disappears  from  A  and  the  entropy  of  A  is  reduced 
without  heat  passing  across  the  boundary  of  A.  But  the 
other  statement  is  still  true :  the  heat  is  given  out  so  as  tc 
cause  at  least  an  equal  simultaneous  increase  of  entropy  else- 
where. There  cannot  be  a  single  thermo- junction  elevating 
heat  into  work  with  no  corresponding  degradation  elsewhere 
in  the  circuit. 


OF  A   BODY.  33 

It  may  be  similarly  shown  that  the  entropy  of  a 
body  can  only  be  increased  reversibly  by  its  taking 
in  heat  through  its  envelope.  Consider  an  isolated 
system  undergoing  a  reversible  change,  so  that  its 
total  entropy  remains  constant  while  the  entropy  of 
one  part  A  is  reduced,  and  that  of  B  increased  by 
the  same  amount.  A  must  give  out  heat,  and  if  B 
increases  its  entropy  without  taking  in  that  heat, 
that  heat  must  go  into  some  third  body  C,  and 
increase  its  entropy  to  at  least  the  extent  that  the 
entropy  of  A  fell.  There  would  then  be  an  increase 
of  the  total  entropy  of  the  system,  so  that  the 
change  is  not  reversible.  To  make  the  change 
reversible  B  must  increase  in  entropy  only  in  pro- 
portion to  the  heat  it  takes  through  its  envelope.* 
It  must  not  be  for  a  moment  supposed  that  because 
the  entropy  of  a  body  can  only  be  increased 
reversibly  by  heat  coming  from  outside  through  the 
envelope,  it  cannot  be  increased  irreversibly  other- 
wise. The  entropy  of  a  body  cannot  be  decreased 
without  giving  out  heat,  either  reversibly  or  irre- 
versibly, and  it  cannot  be  increased  reversibly 

*  The  thermo- junction  case  is  omitted  from  consideration 
here.  B  could  in  theory  increase  its  entropy  reversibly  by 
means  of  a  thermo -contact  inside  it ;  but  even  then,  though 
its  heat  is  not  taken  in  as  heat  through  its  envelope,  its 
increase  of  entropy,  in  a  reversible  change,  is  exactly 
balanced  by  the  decrease  of  entropy  of  another  body  which 
does  not  lose  the  corresponding  heat  through  its  envelope  as 
heat. 

E.  D 


34  ENTROPY. 

without  taking  in  heat;  but  it  can  be  increased 
irreversibly  without  taking  in  the  corresponding 
heat  through  its  envelope. 

As  the  entropy  of  a  body  is  decreased  by  giving 
out  heat  only,  its  entropy,  compared  with  that  of 
the  same  body  in  the  standard  state  can  be  found 
by  bringing  it  to  the  standard  state  by  a  reversible 
change  or  a  series  of  reversible  changes,  and  finding 
the  heat  given  out  and  the  temperature  at  which  it 
is  given  out,  and  the  corresponding  entropy.  If  the 
heat  H  can  all  be  given  out  at  one  temperature  0, 
then  H/0  is  the  entropy,  taking  that  of  the  standard 
state  as  zero.  If  the  heat  is  given  out  at  various 
temperatures,  the  entropy  isfdR/0.  If  the  entropy 
of  the  body  is  less  than  in  the  standard  state,  as  in 
the  case  of  a  block  of  ice  at  its  melting-point,  the 
entropy  can  be  measured  by  finding  the  heat  taken 
in  during  a  reversible  change  to  the  standard  state, 
and  giving  it  the  negative  sign. 

Definition. — We  may  therefore  define  the  entropy 
of  a  body  in  state  B,  compared  with  its  standard 
state  A,  as  being  numerically  equal  to  the  heat  that 
would  have  to  be  taken  in  to  get  it  from  A  to  B  by 
reversible  changes,  divided  by  the  absolute  tempera- 
ture ;  or  H/0,  if  the  heat  is  taken  in  (or  given  out 
if  the  entropy  is  less  than  in  state  A)  at  constant 
temperature  0,  or  fdH/0  if  the  temperature  varies. 
Thus  the  entropy  of  the  body  in  state  B  is  not  a 
function  of  the  heat  actually  taken  in  during  its 


DEFINITION.  35 

change  from  A  to  B,  as  the  change  must  have  been 
partially,  and  may  have  been  wholly,  irreversible ; 
but  it  can  be  measured  as  a  function  of  the  heat 
which  would  have  to  be  taken  in  to  change  from  A 
to  B  reversibly,  or  which  would  have  to  be  given 
out  if  the  substance  were  changed  from  B  to  A 
reversibly,  which  amounts  to  the  same  thing. 

It  is,  perhaps,  not  unnecessary  to  point  out 
again  that  though  the  entropy  is  measured  in 
terms  of  heat  that  would  have  to  be  taken  in,  and 
the  temperature  at  which  it  would  come  in  in 
changing  from  A  to  B  reversibly,  the  entropy  is, 
in  fact,  always  greater  than  H/0  or  J  dH/0,  as  a 
reversible  change  never  takes  place.  In  a  real 
change  3>  >  J  dYLjO ;  and  while  4>  is  positive, 
fdH/0  may  even  be  negative — that  is  to  say,  the 
body  may  change  from  state  A  to  state  B,  increasing 
its  entropy  all  the  time,  and  giving  out  heat  all  the 
time  too.  As  every  text-book  on  the  steam-  and 
gas-engine  (except  Kankine's*)  defines  entropy  as 
fdH/O  instead  of  as  being  measured  by  fdH/0, 

*  Rankine  is  not  clear  about  his  ' '  thermodynamic  func- 
tion." His  first  definition  makes  it  the  same  as  entropy,  but 
lie  also  makes  it  equal  tofdIL/0  without  limiting  the  change 
to  reversible  change.  He  certainly  did  not  develope  the  idea 
of  entropy  and  its  relation  to  waste,  which  forms  the  basis  of 
this  book.  No  doubt  a  man  of  his  ability,  if  he  had  written 
on  steam-engines  somewhat  later,  would  have  been  not  only 
perfectly  correct,  but  also  clear  and  unambiguous  in  his 
statements  and  definitions. 

D2 


36  ENTKOPY. 

if  the  change  is  imagined  by  a  reversible  path  from  A 
to  B,  this  point  cannot  be  insisted  on  too  strongly. 
The  incorrect  definition  makes  the  conception  of 
entropy  impossible  to  the  reader  ;  and  if  the  writer 
did  not  unconsciously  depart  from  his  own  definition, 
it  would  make  nonsense  of  thermodynamics. 

As  the  entropy  of  a  bod}7  can  be  measured  by 
bringing  it  back  to  the  standard  state  by  rever- 
sible changes — that  is  to  say,  changes  which 
involve  no  waste,  and  therefore  no  increase  of 
entropy  in  the  body  and  its  surroundings — the 
entropy  in  state  B,  compared  with  state  A,  is 
measured  by  seeing  how  much  entropy  comes  out 
on  changing  it  from  B  to  A  without  creating  any 
entropy  during  the  change. 

The  entropy  of  a  body  in  a  given  state,  therefore, 
depends  only  on  the  state,  and  not  on  its  past 
histor}r ;  thus  a  quantity  of  steam  in  a  given  state  as 
to  pressure,  temperature,  and  volume  must  give  out 
the  same  entropy  on  being  brought  by  reversible 
change  to  the  state  of  ice-cold  water,  in  whatever 
wa}r  it  was  made  into  steam. 


CHAPTER  III. 

THE    6<f>    DIAGKAM. 

WE  may  now  deal  with  the  0  <£  diagram,  taking  a 
simple  case  to  start  with,  and  discussing  it  from 
the  point  of  view  put  forward  in  this  article. 

We  may  begin  with  the  case  of  a  perfect  gas. 

Suppose  we  start  with  a  quantity,  say,  a  pound, 
so  that  we  deal  with  the  specific  volume  and  entropy 


of  a  perfect  gas  in  a  cylinder  with  heat-tight  sides, 
and  let  the  area  of  the  piston  be  unit}',  so  that  its 
pressure  and  volume  are  given  by  the  point  A  in 
Fig.  4,  where  the  height  is  the  pressure  on  the 
piston,  and  the  horizontal  distance  the  volume  of 
the  gas.  We  may  take  the  state  A  as  standard  for 
simplicity — that  is,  we  take  its  entropy  as  zero. 
In  the  Ocf>  diagram,  to  correspond  with  Fig.  4,  we 


38 


THE  6$   DIAGRAM. 


therefore  start  with  A  to  correspond  with  A  in 
Fig.  4,  marking  off  the  height  O  A  to  correspond 
with  the  temperature  61  of  the  gas  to  start  with,  and 
on  the  line  of  zero  entropy.  Let  the  gas  be  now 
allowed  to  expand,  doing  external  work  MI  and  taking 
in  heat  hlt  at  temperature  Ol  the  expansion  being 
carried  on  reversibly  to  the  volume  and  pressure 
represented  by  B,  Fig.  4. 

Consider  the  position  at  B.     The  gas  has  taken 
in  heat  7^,  but  it  is  no  hotter ;  that  is  to  say,  some- 


thing external,  say  a  heat  reservoir,  has  given  out 
heat  hl9  which  has  crossed  the  boundary  or  envelope 
of  the  gas,  and  has  then  ceased  to  exist  as  heat, 
being  converted  into  work  ?r,  which  has  gone  outside, 
say,  into  a  flywheel.  But  as  the  change  is  rever- 
sible, so  that  the  entropy  of  the  gas  and  the  reser- 
voir is  constant,  and  as  the  reservoir  has  lost  heat 
7tx  at  temperature  0P  and  the  gas  has  taken  in  heat  7^ 
at  temperature  6l9  and  has  given  out  no  heat  else- 
where, its  entropy  must  have  increased  by  Ii\j6\» 
In  Fig.  5,  from  A  draw  A  B,  so  that  A  B 


DIAGRAM  FOB   GAS. 


39 


represents  MVl9  or  <£.  The  height  representing  the 
temperature'  is  constant,  so  A  B  is  a  horizontal 
straight  line. 

That  the  entropy  of  something  outside,  which 
supplied  the  heat  hl  reversibty,  diminished,  may 
seem  an  insufficient  reason  for  saying  the  entropy 
of  the  gas  is  increased.  But  the  gas  cannot  be 
compressed  to  its  old  state  without  giving  out  heat 
at  some  temperature,  of  which  part  must  be  waste. 


The  next  step  is  to  expand  the  gas  from  B  to  C 
reversibly,  without  taking  any  heat,  but  doing 
external  work,  so  that  the  heat  of  the  gas  is  drawn 
upon  to  do  the  external  work.  The  entropy  remains 
constant  during  the  change,  as  the  process  is  rever- 
sible. It  could  not  reduce  the  entropy  during  this 
change,  as  no  heat  comes  out  across  the  envelope, 
and  that  is  the  only  way  the  entropy  could  be 
reduced.  As  the  change  from  B  to  C  is  reversible 
by  hypothesis,  the  entropy  of  the  gas  cannot  have 


40 


THE   0<l>  DIAGRAM. 


increased,  else  it  would  have  to  give  out  heat  on 
reversal  to  get  back  to  the  state  of  B.  From  B  to 
C  then  the  temperature  falls  to,  say,  02,  while  the 
entropy  remains  constant.  From  B  on  Fig.  5  draw 
B  C,  so  that  the  height  of  C  represents  02,  the 
entropy  D  C  being  equal  to  A  B.  Let  the  gas  be 
compressed  isothermally  and  reversibly  from  C  to 
D,  giving  out  heat  /i2  at  temperature  0%  until  the 

G 


c 


entropy  is  reduced  to    zero — that    is    to    say,  its 
original  value. 

On  Fig.  5,  the  state  point  therefore  moves  along 
C  D  to  D,  where  the  temperature  is  02  and  the 
entropy  zero,  as  at  A.  The  gas  is  then  compressed 
reversibly  without  any  passage  of  heat  across  the 
boundary  surface,  until  the  gas  reaches  its  old 
temperature  Olf  If  the  temperature  is  6lt  the  new 
state  point  must  be  on  the  curve  A  B,  and  as  the 
entropy  is  zero,  the  new  point  must  be  A,  which  is 
the  original  starting  point.  We  have  thus  gone 


6+  DIAGRAM   FOR   GAS. 


41 


through  the  ordinary  Carnot  cycle,  and  Fig.  4  is 
the  p  v  diagram,  and  Fig.  5  the  6  <f>  diagram  of  it. 
As  the  cycle  described  is  reversible,  the  curve  A  B, 
Fig.  4,  is  for  a  perfect  gas,  p  v  =  R  0,  where,  as 
the  expansion  is  isothermal,  0  is  constant,  and 
R  is  constant  anyhow,  so  the  curve  is  part  of 
rectangular  hyperbola,  and  is  easily  described  with 
the  help  of  a  slide-rule.  For  instance,  if  R  0  is 
found  on  one  of  the  scales,  and  the  slide  turned 
round,  and  the  1  of  the  same  scale  put  opposite 


0,««;  V        6'  H  J 

R  6,  the  corresponding  values  of  p  and  v  are  read 
off  without  moving  the  slide.  As  the  height  is  equal 
to  the  pressure  on  the  piston,  and  the  length  to  the 
volume,  the  area  A  B  H  F  is  equal  to  w\,  the  external 
work  done  during  expansion ;  and  the  area  is  also 
proportional  to  hlt  the  heat  taken  in  at  tempera- 
ture BI.  In  the  B  <£  diagram,  as  the  change  from  A 
to  B  is  reversible,  and  at  constant  temperature  019 
the  entropy,  &i/#i,  multiplied  by  019  is  obviously 
equal  to  the  heat  hl9  so  the  area  A  B  H  0  (Fig.  5) 
is  also,  in  this  case,  equal  to  the  heat  taken  in 


42  THE    00   DIAGRAM. 

and  to  the  work  done.  It  is  probably  this  that 
has  misled  people  into  supposing  the  6<j>  is  a 
heat  diagram,  or  that  its^^rea  is  proportional  to 
some  existing  heat  or  to  some  work.  Suppose  the 
expansion  from  A  to  B  had  been  carried  out  irrever- 
sibly without  any  heat  be4ng  taken  in  by  the  gas, 
and  without  doing  any  external  work.,  For  instance, 
a  partition  might  be  put  across  the  C3Tlinder  at  A,  to 
hold  the  gas,  while  the  piston  is  moved  to  B.  If  a 
hole  is  now  made  in  the  partition,  the  gas  will 
blow  through  into  the  vacuum,  and  eventually  the 
gas  will  be,  according  to  Joule's  experiment,  at  the 
same  temperature  as  before,  and  therefore  at  the 
same  pressure,  as  the  volume  is  the  same.  The 
gas  has  therefore  got  to  state  B  without  giving  out 
any  work,  and  without  taking  in  any  heat.  It 
must  be  remembered  that  F  A  and  H  B  represent 
the  pressures  on  the  piston,  so  that  in  the  last  case 
A  B  H  F  was  equal  to  the  external  work  7^.  In 
this  case  there  was  no  pressure  on  the  piston 
during  its  movement.  The  passage  from  A  to  B 
was  really  not  by  the  curve  that  joins  them,  but 
from  A  down  to  F,  then  to  H,  and  up  to  B.  There 
is  apt  to  be  some  confusion  as  to  what  is  meant  by 
the  pressure  of  the  gas  during  such  an  irreversible 
change  ;  but  the  only  pressure  that  has  to  do  with 
external  work,  which  is  the  matter  in  question, 
on  an  indicator  diagram,  such  as  Fig.  4,  is  the 
pressure  on  the  piston.  We  might  have  had  an 


0<t>  DIAGRAM   FOR   GAS. 


43 


indicator  on  the  cylinder  which  would,  if  put  on  the 
gas  side  of  the  partition  we  inserted,  have  read 
pressure  as  F  A  throughout  the  movement  of  the 
piston.  In  a  steam-  or  gas-engine  the  indicator  is 
merely  to  tell  the  piston  pressure.  An  indicator 
put  on  this  cylinder,  unless  put  on  the  piston  side 
of  the  division,  would  only  give  incorrect  infor- 
mation. In  a  p  v  diagram  we  are  only  concerned 
with  the  real  piston  pressure,  not  with  any  state  of 
the  gas. 


If  instead  of  using  a  partition,  and  opening  a 
hole  to  let  the  gas  through  after  the  movement  of 
the  piston,  we  had  used  a  very  light  piston  and 
moved  it  suddenly  so  as  to  allow  the  gas  to  expand 
freely  from  A  to  B,  immediately  after  the  move- 
ment of  the  piston,  we  can  imagine  the  piston 
moved  so  suddenly  and  quickly  that  during  the 
movement  there  is  no  pressure  on  it,  and  therefore 
no  external  work  is  done.  The  gas  then  rushes 
into  the  vacant  space,  and  the  work  of  expansion 
of  each  of  the  little  parts  of  the  gas  is  spent  on 


44  THE   0$   DIAGRAM. 

imparting  kinetic  energy  to  other  parts  of  the 
gas.  This  is  degraded  again  into  heat,  and  the 
final  result  is  that  when  the  gas  settles  down  to  a 
uniform  temperature  and  loses  its  kinetic  energy, 
the  pressure  on  the  piston  is  H  B,  and  the  tempera- 
ture Ol9  and  the  entropy  A  B.  It  has  been  urged 
by  men  of  science  too,  that  in  this  case  the  increase 
of  entropy  is  due  to  the  production  of  this  heat  by 
friction  in  wiping  out  the  kinetic  energy ;  and  in 
support  of  this  contention  it  must  be  noted  that  if 
the  gas  puts  all  its  work  of  expansion  into  kinetic 
energy,  and  that  is  then  degraded  into  heat,  the 
heat  so  produced  will  be  equal  to  the  heat  /ix  that 
would  have  been  taken  in  during  the  expansion  if 
it  had  been  reversible  and  isothermal.  But  the 
whole  of  the  heat  that  is  produced  by  friction  in 
the  reduction  of  kinetic  energy  in  the  irreversible 
expansion  has  previously  disappeared  from  the  gas 
in  producing  the  kinetic  energy.  If  the  heat  pro- 
duced inside  the  gas  by  friction  is  to  be  counted  as 
generated,  the  equal  heat  lost  in  producing  the 
kinetic  energy  must  also  be  counted,  and  the  total 
internal  generation  of  heat  is  then  zero. 

It  has  been  urged  that  the  increase  of  entropy  in 
irreversible  expansion  is  still  due  to  the  gas  receiv- 
ing heat,  and  the  increase  of  entropy  is  still  fdh/0, 
even  in  an  irreversible  change.  This  is  due  to  a 
misunderstanding  of  the  meaning  of  the  symbols. 
h  is  not  the  heat  of  the  substance,  nor  the  heat 


0<f>  DIAGRAM   FOR   GAS.  45 

gained  by  the  substance.  It  is  the  heat  taken  in  by 
the  substance  from  the  outside.  It  is  the  loss  of 
heat  suffered  by  some  external  reservoir  or  body. 
If  you  give  h  a  different  meaning,  so  as  to  include 
heat  generated  by  any  means  inside  the  body  with- 
out coming  through  the  envelope  or  case  as  heat,  h 
increases  when  a  gas  is  compressed  adiabatically — 
that  is  to  say,  without  any  heat  passing  through  the 
envelope — and  decreases  during  adiabatic  expansion. 
And  if  the  equation  f  d]i\Q  =  $  were  still  true  when 
dh  meant  the  increase  of  heat  of  the  body,  the 
reversible  adiabatic  compression  of  a  gas  would  not 
be  isentropic.  On  the  other  hand,  the  isothermal 
reversible  expansion  of  a  perfect  gas  would  be 
isentropic,  because  the  heat  taken  in  would  be 
exactly  balanced  by  the  internal  disappearance  of 
heat  which  is  converted  into  external  work.  It  is 
important  to  remember  that  in  these  equations  h,  or 
H  stands  for  the  heat  passed  into  the  working 
substance,  from  outside  through  the  envelope,  and 
not  for  the  heat  of  the  substance,  or  for  heat 
generated  inside  by  friction,  by  internal  combustion, 
by  an  electric  resistance  worked  from  outside,  by 
compression  or  otherwise. 

It  may  be  asked  what  meaning  the  curve  A  B, 
Fig.  4,  has  when  the  expansion  is  quite  irreversible 
— that  is  to  say,  all  the  increase  of  entropy  of  the 
substance  is  uncompensated  by  reduction  of  entropy 
elsewhere,  and  there  is  no  external  work.  If  the 


46 


THE  *+  DIAGRAM. 


piston  is  moved  suddenly  to  B,  after  a  little,  the 
temperature,  which  varied  throughout  the  volume  of 


the  gas  during  the  rush,  comes  to  $  again,  and  the 
pressure  to  H  B. 

If  the  piston  were  first  moved  suddenly  to  a  point, 
saj,  half  way  along,  and  stopped,  the  pressure  would 
correspond  to  the  vertical  height,  say  K  E,  Fig.  6. 


The  piston  might  divide  its  whole  stroke  into  a 
series  of  little  quick  steps,  each  being  so  quick  that 
the  gas  exerts  no  pressure  on  the  piston  during  its 
motion.  The  diagram  win  then  be  a  sort  of  comb 
of  vertical  lines,  the  tops  being  in  the  curve  A  B, 


*4>  DIAGRAM  FOR  GAS. 


and  there  will  be  no  area  and  no  outside  work.  If 
these  steps  are  made  more  numerous  and  smaller, 
there  will  still  be  no  area,  enclosed  «nd  no  external 
work.  The  «malW  each  step  is  made,  the  «"»«1W 
will  be  the  variation  of  temperature  throughout  the 
during  the  little  rashes  of  giy,  and 


tion  of  pressure  i 
of  the  cylinder. 
If  the  jumps  are 


my, 


the 


indefinitely  <anall1  the 


of  the 

the  pressure  on  the 
remains  constantly  zero,  so 
done,  and  there  is  no  area 
of  the  gas  on  the  walk  of  the  cylinder  follows  the 
OTTO  A  B  (Fig.  4)  as  if  the  expansion  we 
and  iftfttJifi  ""*! 

Turning  to  the  0+  diagram  (Fig.  5), 
lessoning  apply  ?    At  B  the  temperature  is 
the  entropy  A  B;  bat  the 


48 


THE   6$  DIAGRAM. 


state  point  come  along  the  straight  line  A  B,  or 
does  it  come  round  by  O  and  H,  so  that  110  area  is 
enclosed  ?  or  does  it  do  something  between  the  two  ? 
To  come  round  by  O  and  II  would  mean  that  the 
gas  had  no  temperature,  or  was  at  absolute  zero 
during  the  change,  which  is  absurd.  If  the  whole 
energy  of  expansion  were  first  devoted  to  producing 
kinetic  energy  of  the  gas,  equal  to  the  external  work 


that  might  have  been  done  by  isothermal  expansion, 
the  temperature  would  fall  to  #2;  and  as  the  kinetic 
energy  became  degraded  into  heat,  there  would  be 
an  increase  of  temperature  and  entropy,  so  that  the 
state  point  would  move  by  a  sloping  curve  from  D 
to  B.  If  the  piston  only  makes  small  jumps,  the 
state  point  will  move  much  as  in  Fig.  7.  The 
temperature  will  fall  a  little.  It  will  be  different 
throughout  the  mass  of  gas,  but  in  no  part  will  it 


6^  DIAGRAM   FOR   GAS.  49 

fall  much.  If  the  expansion  takes  place  in  a  succes- 
sion of  jumps,  the  state  point  will  make  a  succession 
of  saw  teeth  ;  and  if  the  jumps  are  indefinitely  small, 
the  state  point  moves  along  the  line  A  B,  tracing 
one  side  of  the  area  A  B  H  O. 

We  are  justified,  therefore,  in  drawing  A  B 
straight  as  representing  the  movement  of  the  state 
point  in  the  $<}>  diagram ;  but  this  long  explanation 
has  been  entered  into  rather  to  clear  up  difficulties 
than  anything  else.  The  area  of  the  0$  diagram  is 
not  energy,  but  the  area  is  important,  for  the  excess 
of  the  area  of  the  0  <f>  diagram  over  the  area  of  the 
p  v  shows  the  badness  of  the  engine  from  a  thermo- 
dynamical  point  of  view. 

There  is  considerable  confusion  often  as  to  the 
meaning  of  a  p  v  diagram  ;  that  is  to  say,  as  tojyhat 
p  means  in  an  irreversible  change.  As  a  p  v  or 
Watt  diagram  is  to  tell  the  work  given  out  by  the 
expansion,  p  is  the  pressure  on  the  piston ;  so  that 
the  external  work  is  w  =  fp  dv.  In  any  reversible 
expansion  the  curve  is  also  a  diagram  of  the  state  of 
the  gas ;  and  people  have  been  misled  into  supposing 
that  the  p  v  diagram  refers  to  the  state  of  the  gas  in 
an  irreversible  change.  This  is  the  usual  confusion 
about  reversible  and  irreversible  changes. 

In  Fig.  6  the  dotted  curve  A  E  B  is  the  diagram 
of  the  equivalent  state  of  the  gas,  and  if  p  is  taken 
as  the  height  of  this  curve  at  any  volumey^p  dv>w, 
and  the  area  does  not  represent  external  work. 

E.  B 


50 


THE    6$  DIAGRAM. 


Also  dh<^p  dv.  There  are,  in  fact,  two  p  v  dia- 
grams— one  in  which  p  is  the  pressure  on  the 
piston,  which  is  not  a  diagram  of  the  state  of  the 
gas  at  all,  and  another  which  is  unconsciously  used 
instead  of  it,  which  gives  the  equivalent  state  of  the 
gas,  but  not  the  external  work.  In  reversible 
changes  the  diagrams  coincide  ;  in  irreversible  they 
do  not,  and  we  have  two  sets  of  pressures.  Writers, 
for  instance,  often  state  that  dh/0  —  d<j>,  and  that 


dh/0  and  d<f>  are  both  complete  differentials  ;  dh/0  is 
not  a  complete  differential  in  terms  of  the  p  and  v 
of  the  diagram  of  the  state  of  the  gas  in  which 
pv  =  RO;  rf<£,  on  the  other  hand,  is  a  complete 
differential  in  terms  of  the  ordinates  of  the  state 
diagram  in  which  p  v  =  R  0,  but  it  is  not  a  complete 
differential  with  reference  to  the  external  work  or 
piston  co-ordinates  of  the  Watt  diagram.  dhlO  =  d<{> 
would  be  true  in  the  ideal  case  of  reversibility  when 
the  two  diagrams  coincide. 

As  the  change  of  entropy  follows  the  state  of  the 


DIAGEA 


51 


gas,  and  not  the  external  work,  the  0  </>  diagram  is 
the  same  as  before.  At  B  the  entropy  is  A  B 
(Fig.  5),  for  at  B  the  gas  is  in  a  given  state,  as  to 
pressure,  temperature,  and  volume,  and,  as  already 
fully  explained,  the  entropy  depends  on  the  state  of 
the  gas,  not  on  how  it  got  into  that  state. 

It  is  therefore  incorrect  to  say  that  the  area 
A  B  H  O,  Fig.  5,  represents  heat,  for  no  heat  was 
taken  in.  It  is  equally  incorrect  to  say  it  represents 

Ilg.6. 


external  work,  for  no  external  work  was  done. 
Thus  entropy  is  not  a  factor  of  heat ;  it  is  not  heat- 
weight,  and  it  is  not  a  factor  of  energy.  It  is  not 
hi/tii,  for  no  heat  has  been  taken  in,  and  the 
temperature  has  been  constant.  Neither  is  it  equal 
tofdhi/9.  It  is,  to  repeat,  a  quantity  which,  when 
multiplied  by  the  lowest  temperature  available,  gives 
the  incurred  waste.  Thus,  if,  for  example,  6%  is 
taken  as  the  lowest  available  temperature,  the  gas 
may  be  expanded  reversibly  without  change  of 

E2 


52 


THE   6$  DIAGRAM. 


entropy  from  B  to  C,  and  may  then  be  compressed 
isothermally  along  C  D,  Figs.  4  and  5,  giving  out 
the  heat  /*2  at  temperature  02.  This  is  represented 
by  the  area  D  C  J  G,  Fig.  4,  and  D  C  H  0,  Fig.  5. 
The  area  D  C  H  0  is  thus  the  waste  that  must  at 
least  result  from  bringing  the  substance  from  a  state 
where  its  entropy  is  A  B  or  C  D  back  to  the 
standard  or  zero  state. 

The  waste  has  not  yet  taken  place  when  the  gas 
has  got  to  B,  even  irreversibly  ;  but  to  get  it  to  A 


Fly.  4. 


again,  work  must  be  degraded  into  heat,  of  which 
O  D,  the  lowest  available  temperature,  multiplied 
by  the  entropy  A  B,  gives  the  waste  D  C  H  O.  That 
is  why  the  expression  "  incurred  waste  "  has  been 
used.  The  waste  does  not  take  place  necessarily 
when  the  entropy  increases  ;  but  as  it  must  take 
place  eventually,  it  is  called  "  incurred."  There  is 
still  another  reason  for  the  expression.  In  reality 
reversibility  is  unattainable,  so  in  bringing  the  gas 
back  from  B  to  A  a  little  more  waste  must  result. 
But  this  waste  is  not  "  incurred  "  when  the  gas  is 
at  B ;  it  is  a  subsequent  product,  due  to  the 


DIAGRAM   FOR   GAS. 


53 


difficulty  of  approaching  reversibility  in  the  changes 
necessary  to  bring  the  gas  back  to  A. 

The  other  definitions  given  also  fit  the  case. 

The  entropy  of  the  gas  at  B  is  numerically  equal 
to  the  heat  that  must  be  given  out  on  returning  to 
A  by  reversible  changes,  divided  by  the  tempera- 
ture of  the  surface  at  which  the  heat  is  given  out.* 
Thus,  if  the  gas  is  brought  back  isothermally  along 
B  A,  Fig.  4,  the  heat  given  out  at  temperature  #1  is 


proportional  to  the  area  A  B  H  0,  Fig.  5,  and  that 
divided  by  O  A  is  A  B.  The  same  reasoning  applies 
to  the  path  B  C  D  A,  Fig.  4,  where  the  heat  H2  or 
D  C  H  0,  Fig.  5,  is  given  out. 

*  In  this  sense  entropy  might  be  called  a  factor  of  heat,  but 
that  is  not  at  all  the  sense  in  which  it  is  so  termed.  If  a 
body  has  entropy  *  to  get  it  the  standard  state  where  *=0, 
heat  H,  so  that  H/0= *  or/r!H/0=*  must  at  least  be  given 
out,  but  that  heat  may  have  to  come  from  external  work, 
and  the  entropy  of  the  body  is  not  a  factor  of  some  heat  of 
unknown  amount  that  is  going  to  be  .produced  at  a  future 
date  from  some  external  energy  or  other. 


54 


THE    6$   DIAGRAM. 


The  curve  B  C,  Fig.  4,  may.be  easily  drawn  with 
the  help  of  a  slide-rule  and  proportional  compasses. 
Its  equation  is  p  v?  =  constant.  This  is  shown  in 
the  text-hooks,  so  it  need  not  be  gone  over  again,  as 


the  object  of  this  book  is  rather  to  be  supplementary 
and  explanator}',  and  not  a  complete  self-contained 
treatise  on  elementary  thermodynamics. 

As  we  know  the  value  of  p  v  at  B,  we  can  find  the 

JKf.fi. 


Q  "-«»         Jf         LK  € 

value  of  p  itf  there,  if  we  know  y.  For  a  perfect 
gas,  7  is  1.6  ;  we  therefore  find  p  vY.  Starting  from 
B,  Fig.  2,  for  instance,  we  multiply  the  corre- 
sponding p  by  the  corresponding  vY,  and  note  the 


6$  DIAGRAM   FOR   GAS.  55 

number  on  the  slide-rule  scale.  The  proportional 
compass  is  set  at  1.6.  For  every  pressure  the 
remainder  of  the  scale  up  to  pv7  is  spanned 
by  the  long  legs  of  the  compass  ;  the  short  legs  then 
give  the  corresponding  v  on  the  same  scale. 

The  word  "  adiabatic  "  means  without  passing 
through,  denoting  that  during  an  adiabatic  change 
there  is  no  crossing  of  heat  into  or  out  of  the  gas. 
The  heat  of  the  gas  may  vary,  of  course,  but  by 
changing  to  or  from  work.  If  it  were  reversible, 
an  adiabatic  would  be  an  isentropic  change.  But 
the  entropy  can  increase  without  any  heat  coming 
in,  and,  in  fact,  always  does  ;  so  an  adiabatic  change 
never  is,  in  fact,  isentropic. 

Text-books  frequently  treat  isentropic  and  adia- 
batic as  synonymous  terms — a  very  widespread 
error,  arising,  like  the  others,  from  ignoring  real  as 
opposed  to  hypothetical  thermodynamics. 

It  will  be  noticed,  on  comparing  the  6<f>  and  the 
p  v  diagram  for  the  Carnot  cycle  of  Figs.  4  and  5, 
that  if  the  diagrams  are  drawn  to  suitable  scale, 
while  the  work  point  in  the  p  v  diagram  traces  the 
first  step  from  A  to  B,  the  state  point  on  the  6  </> 
traces  A  B.  The  area  A  B  H  F  on  Fig.  4  is  then 
equal  to  the  heat  hi  taken  in  at  0,  and  to  the  external 
work  done  from  A  to  B.  The  heat  of  the  perfect 
gas  remains  constant.  The  corresponding  area, 
A  B  H  O,.Fig.  5  is  also  equal  to  the  heat  hi  taken 
in  at  temperature  fy.  But  the  area  A  B  H  O  does 


56 


THE    6$   DIAGEAM. 


not  represent  the  increase  of  heat  of  the  gas.  The 
diagram  is  not  a  heat  diagram,  in  fact.  From  B  to  C 
the  area  B  C  J  H  represents  the  work  done  l>y  the 
gas,  but  there  is  no  heat  taken  in  or  given  out ;  but 
Fig.  s. 


the  gas  loses  heat,  which  goes  out  as  work.  The 
state  point  in  the  0<j>  diagram  merely  descends. 
The  other  two  steps  are  similar,  but  in  the  opposite 
direction.  The  result  is  that  the  area  A  B  C  D  is 


equal  on  these  diagrams.  The  efficiency  in  Fig.  4 
is  the  ratio  of  the  area  A  B  C  D  to  the  area  A  B  H  F. 
The  efficiency  in  Fig.  5  is  the  ratio  of  the  area 
ABCD  to  ABFO,  and  is  much  easier  to  see, 


6$   DIAGRAM    FOR   STEAM.  57 

and  is  obviously  proportional  to  (#1  -  O^/Oi.  But 
that  is  only  true  of  the  reversible  cycle  ;  if  the  cycle 
is  irreversible,  for  instance,  on  account  of  the 
expansion  from  A  to  B  being  free,  Fig.  5  does 
not  give  the  efficiency.  It  is  worthy  of  remark 
that  in  the  reversible  cycle  the  entropy  alters 
during  the  changes  in  which  the  heat  of  the  gas 
remains  constant,  while  the  entropy  remains  con- 
stant during  steps  two  and  five,  in  which  the  heat 
varies. 

0(f>  Diagram  for  Steam. — Instead  of  taking  a 
perfect  gas,  steam  may  be  taken.  Water  at  the 
temperature  of  ice  is  generally  taken  as  standard  of 
comparison.  The  water-steam  diagram  may  there- 
fore be  started  from  32°  Fahr.,  or  493°  F.A.,  or 
Fahrenheit  absolute.  The  unit  of  heat  used  by 
English  engineers  is  the  heat  necessary  to  raise 
1  Ib.  of  water  1°  Fahr.,  namely,  from  59°  to  60°. 
It  is  a  pity  there  is  a  special  unit  of  heat.  It  is  a 
survival  of  the  times  when  people  did  not  fully 
realise  that  heat  is  energy ;  but  there  is  no  reason 
why  we  should  still  have  a  separate  unit  of  heat. 
But  as  we  have  the  foot-pound  and,  I  think,  the 
poundal,*  as  units  of  energy,  perhaps  a  special 
unit  of  heat  saves  confusion.  The  heat  necessary 
to  raise  1  Ib.  of  water  1°  Fahr.  is  called  the  British 

*  I  think  poundals  are  energy,  but  they  may  be  force,  or 
a  kind  of  yellow  cakes.  I  do  not  remember,  and  am  not 
anxious  to  know. 


58  THE    00   DIAGRAM. 

thermal  unit ;  and  a  quantity  of  heat  is  measured 
in  British  thermal  units.* 

The  Institution  of  Civil  Engineers  has  recently 
recommended  officially  that  "  British  thermal  unit" 
be  shortened  into  B.Th.U.  as  opposed  to  B.T.U., 
which  stands  for  "Board  of  Trade  Unit,"  an  irregular 
and  barbarous  name  of  3,600  joules.  No  doubt 
British  thermodynamicians  are  getting  so  efficient 
in  manipulating  their  units  that  inaccuracies  due 
to  the  interchangeable  use  of  the  British  thermal 
and  the  Board  of  Trade  unit  are  becoming  per- 
ceptible. The  letters  B.Th.U.  will,  therefore,  be 
used  to  denote  these  things. 

To  raise  water  1°  at  a  given  temperature,  or 
sensibly  at  that  temperature,  needs  1  B.Th.U. 

per  lb.,  so  the  increase  of  entropy  per  degree  is  -  or 

~  =  — ;  provided  the  specific  thermal  capacity, 
d9  0 

inaccurately  called  the  "  specific  heat,"  of  the  water 
remains  constant.  The  0$  curve  for  water  is 

*  Lord  Kelvin  and  others  have  pointed  out  that  this  use  of 
the  word  "units"  is  incorrect.  The  guinea,  pennyweight, 
perch,  quartern,  bushel,  acre,  kilderkin,  hand,  baker's  dozen, 
ream,  em,  month  of  Sundays,  and  blue  moon  are  British 
units,  so  that  a  statement  as  to,  say,  31,416  British  units 
would  refer  to  a  sort  of  museum  that  could  easily  be  collected. 
But  it  is  no  use  being  too  pedantic  over  such  things.  What 
is  wanted  is  a  name  for  the  British  thermal  unit,  such  as  the 
therm,  then  we  could  say  that  the  water  took  in  so  many 
therms.  It  would  be  simpler  still  to  say  so  many  foot- 
pounds. 


DIAGRAM   FOE   STEAM. 


59 


thus    sucli     that    the     rate     of    increase    of    the 
horizontal    line   is    inversely   as    the    height.     As 

^  =  I/O,  0  =   fldO  =  loge  0  +   const.      The 
6<f>  for  a  substance  whose  thermal  capacity  remains 


constant  is  thus  a  logarithmic  curve,  and  the  con- 
stant is  chosen  so  that  the  curve  cuts  the  ordinate 
O  P,  Fig.  8,  at  the  height  corresponding  to  493° 
Fahr.  This  curve  can  be  easily  plotted  by  calcula- 
tion, and  more  easily  copied  from  a  steam  table  or 
Sankey's  valuable  steam  chart.  The  steam  chart, 
for  which  engineers  are  deeply  indebted  to  Captain 


60 


THE 


DIAGRAM. 


Sankey,  shows  properties  of  steam  graphically,  being 
a  6<f>  diagram  with  curves  giving  every  possible 
information  carefully  plotted  on  it.  The  statement 
that  a  square  of  a  given  size  is  a  British  thermal 
unit  need  not  be  regarded.* 


It  need  hardly  be  mentioned  that  it  is  much  easier 
to  work  all  thermodynamical  calculations  on  the 
metric  and  especially  the  C.  G.  S.  system  ;  but 
there  seem  to  be  no  convenient  tables  available. 

*  For  accurate  work  a  correct  steam  table  or  chart  is 
necessary.  Probably  the  National  Physical  Laboratory  will 
soon  check  Kegnault's  experimental  figures.  The  present 
steam  tables  are  in  urgent  need  of  careful  revision. 


e$  DIAGRAM   FOR   STEAM.  61 

It  is  one  of  the  strongest  arguments  against  the 
adoption  of  the  metric  system  in  this  country  that 
Continental  people,  even  scientific  people,  who  could 
use  it  without  any  difficulty,  generally  try  to  avoid 
its  use.  A  steam  table,  with  the  temperatures  in 
degrees  from  freezing-point,  the  pressures  in  milli- 
metres of  mercury,  atmospheres,  or  kilogrammes, 
instead  of  megadynes  per  square  centimetre,  the 
heats  in  calories  instead  of  joules,  and  the  entropies 
not  given  at  all,  is  not  convenient;  and  it  is  easier, 
even  if  it  is  desired  to  work  on  the  metric  system,  to 
use  the  English — or  rather  American — steam  tables, 
such  as  Peabody's  or  Reeve's.  Peabody's  and 
Reeve's  can  both  be  got  separately,*  and  Reeve's 
are  especially  complete  and  convenient.  The 
expression  "  Entropy  of  the  Heat-Energy "  at 
the  head  of  the  table  does  not  make  the  data 
inaccurate. 

Starting  from  492.8°  F.A.,  Fig.  8,  the  curve 
during  the  heating  of  the  water  is  practically 
logarithmic.  It  would  be  accurately  logarithmic  if 
the  specific  thermal  capacity,  or  specific  "heat"  of 
the  water  were  constant.  The  curve  is  easily  got 
from  the  temperature  T,  and  entropy  N,  columns,! 

*  Messrs.  Wiley  &  Co.  and  Messrs.  Macmillan  &  Co. 

t  It  seeins  a  pity  to  introduce  a  new  letter,  N,  for  entropy, 
Eankine  used  <p  for  the  thermodynamic  function,  and  Clausius 
used  S  for  entropy.  Maxwell  used  0  and  $  for  temperature 
and  entropy.  Gibbs  used  i\  for  entropy.  T  and  t  are  con- 


62 


THE    6$   DIAGRAM. 


and  as  the  curvature  is  very  small,  a  few  points  are 
enough.  Suppose  the  water  is  heated  up  to  850° 
F.A.,  omitting  decimals,  before  steam  begins  to 
form.  The  table  gives  220  Ib.  per  square  inch  as 
the  pressure,  and  the  volume  is  inappreciable.  The 
indicator  or  p  v  diagram  may  be  drawn  at  the  same 
time  (Fig.  9)  with  the  lettering  to  correspond.  The 
0<£  diagram,  so  far,  is  a  piece  of  a  logarithmic 
curve,  and  at  850°  F.A.  the  entropy  is  0.552.  This 


means  that,  to  get  the  water  back  to  the  standard 
temperature  of  493°  F. A.,  493  X  .552  =  272  British 

venient  for  absolute  and  thermometer  temperatures,  but  T 
is  already  in  use  for  kinetic  energy,  and  this  gives  rise  to 
confusion  in  dealing  with  thermodynamics  and  the  kinetic 
theory  of  gases  together.  The  Germans  often  use  $•  for 
temperature.  Quantity  of  heat  taken  in  by  a  body  is 
generally  written  Q,  but  as  Q,  in  other  branches  of  physics 
denotes  the  quantity  factor  of  energy,  and  not  energy  itself, 
H  is  more  accurate.  0,  <j>,  and  H  are  therefore  used  here, 
following  Maxwell  and  others,  though  T,  S,  and  Q  are  more 
general,  especially  in  scientific  books  and  papers. 


6$   DIAGRAM   FOR   STEAM.  63 

thermal  units  must  be  wasted,  if  493°  F.A.  is  the 
lowest  available  temperature.  It  does  not  follow 
that  this  waste  has  been  caused  by  warming  this 
water.  If  the  0.552  was  compensated  entropy — 
that  is  to  say,  if  the  increase  of  entropy  of  the  water 
was  exactly  balanced  by  an  equal  decrease  of 
entropy  of  whatever  supplied  the  heat — there  was 
no  increase  of  total  entropy,  and  therefore  no  new 
waste  incurred.  If,  on  the  other  hand,  the  water 
was  sensibly  at  the  same  temperature  throughout 
its  volume  at  each  instant,  but  was  heated  by  gases 
at  a  much  higher  temperature,  there  was  a  growth 
of  uncompensated  entropy  outside  the  water  during 
the  heating.  The  growth  of  entropy  due  to  heat 
conduction  will  be  discussed  presently.  In  the  case 
of  heating  water,  practically  all  the  energy  taken 
in,  whether  as  heat,  or  as  work  which  is  degraded 
into  heat  in  the  body  of  the  liquid,  is  in  the  form 
of  sensible  heat,  and  it  might  appear  that  the 
entropy  was  a  factor  of  the  heat ;  for  though^  dh/0 
<  </>,  where  h  is  the  heat  taken  in  as  heat,  if  the 
process  is  not  reversible,  fdu\B  =  0  is  very  nearly 
true  where  duis  the  differential  of  the  sensible  heat 
of  the  body.  It  is  not  quite  true,  however.  The 
water  can  do  external  work  during  the  expansion, 
and  then  even  jdujO  <  <j>.  It  is  quite  clear  that  the 
entropy  of  the  water  at  B — that  is  to  say,  at  850° 
F.A.  is  0.522,  however  the  heating  has  been  carried 
out.  Text-books  on  the  steam-engine  define  the 


64  THE    e$   DIAGRAM. 

entropy  &sfdh/0,  but  if  the  water  is  heated  in  any 
other  way  than  by  taking  in  heat  from  the  outside — 
for  instance,  by  heating  by  Foucault  currents  pro- 
duced in  the  water  by  a  rapidly  changing  electrical 
field — fdh\B  might  be  zero  if  the  water  took  in  no 
heat,  and  negative  if  it  gave  out  some  as  it  heated. 
It  cannot  be  too  often  urged  that  the  entropy  of  a 
body  depends  on  its  condition  and  not  on  its  history. 
Unless  all  the  changes  are  reversible,  fdh\ '6  depends 
on  the  history,  not  on  the  condition  of  the  sub- 
stance. If  the  writers  of  the  text-books  stuck  to  their 
definition  of  entropy  when  working  with  irreversible 
processes,  they  would  come  to  grief  at  once ;  but 
they  unconsciously  avoid  the  abyss  that  ought  to 
yawn  for  them  by  taking  their  data  from  steam 
tables  or  Sarikey's  chart,  which  is  correct,  because 
the  entropy  then  depends  on  the  state  of  the 
substance  and  not  onfdh/0. 

Continuing  the  diagrams,  suppose  all  the  water 
vaporised  at  pressure  220  Ib.  and  temperature  850° 
F.A.  The  p  v  diagram  is  traced  by  the  constant 
pressure  line  B  C  to  2.1  cubic  feet.  The  state 
point  on  the  6$  diagram  goes  to  <£  =  1.537.  At 
the  point  C  the  p  v  diagram  shows  the  external 
work  done  by  the  steam  in  expanding.  It  does  not 
matter  whether  the  steam  is  in  the  boiler  or  the 
cylinder  for  the  moment ;  the  work  is  done  outside 
the  steam  during  expansion.  We  may  assume  the 
steam  to  be  in  a  hypothetical  cylinder.  Part  of  the 


6<t>   DIAGRAM   FOR   STEAM. 


65 


heat  taken  in  is  converted  into  work — namely,  84.8 
British  thermal  units,  and  this  is  shown  on  the  p  v 
diagram.  753  British  thermal  units  have  been 
devoted  to  vaporising  the  water.  This  energy  is 
still  in  the  steam.  The  allocation  of  the  energy 


taken  in  by  the  water  in  evaporating,  which  we  will 
assume  was  taken  in  as  heat,  might  be  shown  on 
0  <f>  curve.  The  area  A  B  B1  0  then  represents  the 
sensible  heat  which  raised  the  temperature  of  the 
water,  or  it  so  nearly  represents  the  area  that  it  may 
be  taken  to  represent  it.  The  variations  of  specific 
thermal  capacity  of  water  are  small.  The  vertical 


66  THE   00   DIAGRAM. 

ordinate  is  temperature,  the  horizontal  entropy. 
Take  now  the  area  B  C  C l  B l ;  it  does  not  represent 
the  increase  of  heat  of  the  water  from  B  to  C,  so  it 
is  not  a  heat  diagram ;  neither  does  it  represent  the 
increase  of  energy  of  the  water  from  B  to  C,  so  it  is 
not  an  energy  diagram  of  the  steam.  It  is  a  6  <f> 
diagram,  and  the  length  B  C,  multiplied  by  the 
lowest  available  temperature,  gives  the  waste.  Also 
the  area  B  C  C1  B1  gives  the  heat  that  must  at 
least  be  given  out  at  temperature  BI  B,  or  850° 
F.A.,  to  get  the  steam  back  into  water  at  the  same 
temperature ;  but  this  heat  would  not  all  be  pro- 
vided by  the  steam;  some  of  it  must  come  from 
outside,  generally  as  mechanical  work.  As  frequently 
explained,  entropy  is  not  a  factor  of  heat.  Suppose 
we  adopt  a  suitable  factor  x>  so  that  0  x  is  *ne 
increase  of  energy  in  the  form  of  heat  in  the  body 
itself  if  the  temperature  has  been  constant,  and 
y#dxis  the  increase  of  heat  from  the  standard  state 
when  the  temperature  varies.  It  must  be  pointed 
out  that  splitting  heat  into  factors  6  and  x>  so  that 
f6  dx  is  energy,  is  unorthodox.  There  has  for  a 
long  time  been  a  sort  of  hazy  idea  in  thermo- 
dynamics that  heat  should  be  split  up  into  a  tension 
and  a  quantity  factor,  like  other  forms  of  energy. 
Taking  temperature  as  one  factor,  capacity  for  heat 
has  been  proposed  as  the  other  ;  but  that  is  obviously 
absurd,  as,  if  C  is  the  capacity,  the  energy  would 
bey  C  dO,  not  J  0  dC,  Besides  C  is  capacity  for 


6$   DIAGRAM   FOR   STEAM.  67 

energy  not  for  a  quantity  factor,  so  that  capacity 
for  heat  is  by  no  means  the  analogue  of  capacity — 
in,  say,  hydraulics  or  electricity.  As  has  been 
frequently  mentioned  already,  the  error  that  entropy 
is  a  factor  of  energy,  is  very  widespread,  not  only 
among  physicists  and  engineers,  but  even  among 
many  who  write  specially  on  thermodynamics.  I 
brought  a  paper  before  the  Physical  Society  about 
two  years  ago,  in  which  it  was  pointed  out  that, 
although  in  a  reversible  change  <j>  is  numerically 
equal  to  the  factor  of  energy — lost  as  heat  by  the 
external  reservoir,  not  gained  as  energy  by  the 
working  substance — corresponding  with  tempera- 
ture, fQ  d<f>  is  not  energy  in  a  real  change.  The 
paper  discussed  the  question  of  splitting  heat  into 
factors  of  energy,  and  compared  the  advantages  and 
drawbacks.  The  paper  was  rejected,  however,  and 
until  scientific  men  generally  attend  to  real  partly 
irreversible  processes,  the  subject  cannot  come 
forward  for  discussion.  It  is  fair  to  point  out  as 
far  as  possible  where  my  treatment  is  not  orthodox, 
so  that  the  reader  does  not  willingly  absorb  without 
criticism,  as  established  thermodynamics,  what  are 
really  the  heterodox  ideas  of  the  writer. 

As  we  have  to  deal  with  sensible  heat,  the  heat 
that  makes  things  hot,  and  latent  heat,  which 
changes  their  physical  state,  for  instance,  the  dif- 
ference of  internal  energy  between  steam  and  water 
at  the  same  temperature,  we  may  use  suffixes,  so 

F2 


68  THE    6$  DIAGRAM. 

that  x*  is  the  quantity  factor  of  sensible  heat,  and 
Xp  that  of  latent  heat  of  physical  change.*  We 
may  now  plot  the  heat  energy  of  the  pound  of  water 
from  A  to  C.  From  A  to  B  we  get  the  figure 
A  B  B1  O,  Fig.  10,  in  which  the  height  is  0  and  the 
breadth  x-?>  so  that  the  area  isfOdxs,  giving  the 
sensible  heat  of  the  water  at  850°  F.A.,  neglecting 
any  question  of  variation  of  specific  capacity.  From 
water  to  steam  at  B  —  850°  the  water  takes  in 
837.7  British  thermal  units  of  energy,  of  which 
753  remain  as  latent  heat,  and  84.7  go  out  as  work. 

*  The  expression  "latent  heat "  is  used  absurdly  in  thermo- 
dynamics. Thus,  if  ice  is  melted,  heat  is  taken  in  and  remains 
in  the  water.  The  alteration  of  volume  is  minute.  If,  on 
the  other  hand,  water  is  evaporated  into  steam  doing  external 
work,  the  external  work  is  included  in  the  so-called  "  latent 
heat."  There  is  the  same  confusion  as  regards  the  mis- 
called "  specific  heat."  The  heat  absorbed  per  unit  mass  at 
constant  volume  may  be  measured  by  a  specific  thermal 
capacity  at  constant  volume;  but  to  measure  the  heat 
absorbed  at  constant  pressure  or  temperature  by  a  ' '  specific 
heat  at  constant  pressure"  or  a  "specific  heat  at  constant 
temperature  "  is  absurd.  In  the  case  of  a  perfect  gas,  for 
instance,  the  difference  between  Kp  and  Kv  is  entirely 
external  work,  and  not  heat  at  all ;  and  the  "  specific  heat  at 
constant  temperature  "  is  entirely  external  work.  The  whole 
nomenclature  of  thermodynamics  demands  re-modelling. 

By  latent  heat  in  connection  with  XP  is  meant  that 
fedxp,  or  generally  e  xp>  is  the  heat  taken  in  and  kept  by  the 
substance  itself  during  a  physical  change  such  as  fusion  or 
evaporation;  and  it  does  not  include  any  external  work 
Similarly  fedxc  is  the  heat  of  chemical  change.  We  are  not 
concerned  with  that  at  present. 


DIAGRAM   FOR   STEAM. 


69 


As  0  is  here  constant,  \p  =  0.9,  and  we  may  add 
a  rectangle  B  C  C1  B1,  to  represent  0  \p>  to  Fig.  10. 
As  this  rectangle  is  the  same  height  as  the  larger 
rectangle  in  the  0  $  diagram,  we  might  cut  an  equal 
area  off  that  area  B  P  P1  B1.  This  area  would 


B'  CJ)' 

represent  the  increase  of  latent  heat  of  the  water 
from  B  to  C,  and  the  remaining  area  P  C  C1  P1 
would  represent  the  external  work.  This  division 
of  the  0  <£  area  into  two  areas  representing  0  xp 
and  w  is  not  really  legitimate.  The  quantities  have 
no  real  place  on  a  0  </>  diagram  ;  and  it  is  only 
because  the  temperature  is  constant  that  they  can 
be  superposed  that  way.  The  diagram  would  look 


70  THE    6$   DIAGRAM. 

as  if  the  latent  heat  had  been  completed  first,  and 
as  if  there  were  a  state  P  in  which  all  the  water 
was  evaporated,  and  no  external  work  yet  done. 
More  than  this,  there  might  have  heen  no  work 
corresponding  to  the  small  area  P  C  C1  P1.  For 
instance,  the  water  might  have  been  evaporated  at 
a  pressure  of  220  Ib.  and  blown  through  a  small 
hole  into  a  space  big  enough  to  hold  all  the  steam 
at  the  same  pressure  and  temperature.  This  pro- 
cess would  be  irreversible,  and  the  area  P  C  C1  P1 
would  represent  external  work  that  might  have 
been  done,  but  was  not.  The  entropy  would  be  the 
same  in  both  cases,  so  would  \p*  The  work  that 
might  have  been  done,  but  was  not,  has  been  called 
"  uncompensated  work,"  I  think,  first  by  Duhem, 
to  correspond  with  the  idea  of  uncompensated 
entropy.  The  term  is  not  happy,  because  the  work 
does  not  exist.  Uncompensated  entropy  is  a  quan- 
tity that  is  produced,  without  the  redeeming  com- 
pensation of  reduction  of  entropy  elsewhere.  Uncom- 
pensated work  is  not  produced  at  all.  Entropy  is  a 
kind  of  drawback,  so  uncompensated  entropy  is  an 
unmitigated  drawback.  But  work  is  the  other  way.  If 
a  British  workman  unit  is  paid  for  work  he  does 
not  do,  it  would  not  express  the  employer's  feelings 
to  say  the  workman  did  uncompensated  work.  It  is 
the  wage  that  is  uncompensated.  It  is  on  such 
grounds  that  I  venture  to  differ  from  so  eminent  an 
authority  as  Duhem.  The  uncompensated  work  is 


6$   DIAGRAM   FOR   STEAM.  71 

thus  the  difference  betweeny#  d<j>  and  h.  "  Uncom- 
pensated  "  work  is  thus  work  that  might  have  been 
gained,  but  was  not.  "  Ungained  work"  might  be 
a  better  term  for  it.  It  is  the  product  of  the 
uncompensated  entropy  and  the  temperature  if  that 
is  constant,  ovfOd<j>u  when  it  is  not,  <£%  being  the 
uncompensated  entropy.  The  ungained  work  at 
any  temperature  6\  is  to  the  corresponding  incurred 
waste  as  0i/#2>  where  02  is  the  lowest  available 
temperature.  Where  a  reversible  change  takes 
place  there  is  no  uncompensated  entropy  and  no 
ungained  work.  The  change  is  then  said  to  be 
made  at  constant  "  thermodynamic  potential " — 
another  rather  unfortunate  term.  The  reversibility 
of  a  change  can  be  expressed  wholly  in  terms  of 
the  co-ordinates  of  the  substance,  such  as  the 
pressure,  temperature,  volume,  entropy,  latent  heat, 
chemical  energy,  &c.,  without  reference  to  the  out- 
side conditions.  These  expressions  were  worked 
out  by  Helmholtz,  Massieu,  and  Gibbs,  and  are 
now  generally  known  as  Gibbs's  functions  or 
thermodynamic  potentials.  If  these  functions  were 
expressed  clearly  in  words,  instead  of  being  treated 
as  obscure  results  of  pages  of  differential  equations, 
they  would  be  quite  useful  to  engineers.  If  the 
common  statement  that  the  area  of  the  0  $  is  the 
same  as,  or  proportional  to  that  of  the  p  v  diagram 
were  correct,  there  would  be  no  such  thing  as 
ungained  work,  and  there  would  be  no  such  thing 


72 


THE    6$   DIAGRAM. 


as  waste,  and  all  steam  and  gas-engines  would  have 
an  efficiency  of  (Q\  —  O^jO^  and  there  would  be  no 
need  to  study  their  thermodynamics,  as  the}r  would 
be  incapable  of  improvement. 

Let    the    steam    now   be   superheated    to,    say, 


1,000°  F.A.  at  the  same  pressure,  and  reversibly. 
The  p  v  diagram  shows  a  corresponding  increase  of 
volume,  and  the  state  point  rises  to  D.  A  new  area 
under  C  D  is  thus  added  to  the  0  (j>  diagram.  The 
added  heat  is  thus  partly  converted  into  external 
work,  and  partly  converted  into  sensible  heat, 
making  the  steam  hotter.  We  may  add  a  curve 


0<i>   DIAGRAM  FOB   STEAM. 


73 


G  D  to  the  6  x  diagram,  Fig.  10,  so  that  the  height 
is  6  and  the  breadth  x*  and  the  area  /^  d\s  is  the 
increase  of  sensible  heat  during  the  superheating. 
The  area  under  C  D,  Fig.  10,  is  less  than  that 
under  C  D  in  the  0  <£  diagram,  Fig.  8,  because  the 
breadth  of  the  heat  diagram  is  x,  the  factor  of  heat, 


and  of  the  0  <f>,  <£  the  entropy,  which  is  always,  in 
fact,  greater.  The  difference  in  area  between  the 
heat  diagram  and  the  0  <j>  diagram  is  thus  the 
external  work  if  the  change  has  been  reversible,  and 
the  external  and  "  uncompensated  "  or  "  ungained  " 
work  if  it  has  been  wholly  or  partly  irreversible. 
What  was  said  about  its  not  being  legitimate  to 


74  THE    6$   DIAGRAM. 

divide  the  area  B  C  C1  B1  of  the  0  <}>  diagram  into 
two  areas  B  P  P1  B1  and  P  C  C1  P1,  one  represent- 
ing heat  and  the  other  work,  is  now  clearer.  The 
6  $  area  under  C  D,  Fig.  8,  with  its  curved  top, 
cannot  be  divided  into  a  heat  area  C  D  D1  C1  of  the 
heat  diagram,  and  an  external  work  area  by  any 
vertical  line.  In  fact,  the  heat  diagram,  Fig.  10,  is 
itself  unlawful,  for  it  has  two  separate  kinds  of 
heat  on  the  same  area.  Thus,  if  the  steam  did  not 
behave  as  a  perfect  gas  during  heating,  but  stored 
up  some  latent  heat,  the  area  C  D  D1  C1  on  the 
heat  diagram  would  really  represent^  d\s  -\-J  0  d  Xp> 
and  the  area  could  not  be  divided  out  into  latent 
and  sensible  heat.  The  area  of  the  heat  diagram 
(Fig.  10)  above  the  lowest  available  temperature, 
M  S,  is  the  available  energy  of  the  steam  in  relation 
to  the  state  point  M.  That  energy  can  be  got  as 
work  by  going  round  from  D  to  M  by  S  reversibly. 
From  D  suppose  the  steam  to  be  expanded 
isothermally,  but  partly  reversibly  and  partly  irre- 
versibly. If  the  expansion  were  wholly  irreversible, 
no  heat  would  be  taken  in  and  no  work  done,  and 
the  trace  of  the  p  v  diagram  would  be  D  L  K  E  ; 
whereas  if  the  change  were  reversible,  the  trace 
would  be  by  the  bit  of  hyperbola  D  E  shown  dotted. 
If  it  is  partly  reversible,  the  pressure  will  be  less 
than  for  the  hyperbola,  and  it  may  be  as  in  the  full 
curve  D  E.  The  area  on  the  6  <f>  diagram  under 
D  E  is  the  same  whether  the  path  in  the  p  v  is  by 


0$  DIAGRAM   FOR   STEAM.  75 

the  dotted  or  full  line,  and  is  equal  to  the  area 
D  E  K  L  of  the  p  v  diagram,  taking  the  dotted  line, 
and  the  difference  is  the  "ungained"  work.  No 
part  of  the  area  under  D  E  of  the  0  <£  diagram  is 
thus  heat,  assuming  the  superheated  steam  to  be  a 
perfect  gas.  In  reality,  allowing  for  steam  not 
being  a  perfect  gas,  the  increase  of  entropy  D  E 
should  be  a  little  more,  and  the  small  difference 
should  cover  an  area  6  xp  which  should  be  added  to 
the  heat  diagram.  That  may  be  neglected. 

The  superheated  steam  may  now  be  expanded  in 
a  heat-tight  non-conducting  cylinder,  in  a  cylinder 
which  gives  up  heat  to  the  steam  as  it  expands,  or 
in  a  cylinder  that  abstracts  heat.  In  order  to 
make  the  discussion  as  complete  as  possible  we 
may  take  some  of  each.  Then  the  steam  may 
expand  reversibly  doing  the  full  complement  of 
external  work,  or  wholly  irreversibly  doing  no  ex- 
ternal work,  or  partly  irreversibly  doing  some 
external  work.  The  reversible  and  irreversible 
expansion  of  a  perfect  gas  have  already  been 
discussed. 

If  the  steam  now  expanded  without  doing  any 
external  work,  its  state  point  would  move  to  the 
right  of  E  on  the  0  <j>  diagram,  and  the  p  v  diagram 
would  go  down  E  K,  and  then  the  volume  would 
increase,  at  zero  pressure.  If  the  cylinder  gave  up 
enough  heat  to  keep  the  steam  at  the  constant 
temperature  of  1,000°  F.A.,  and  the  expansion  was 


76 


THE    6<l>   DIAGRAM. 


reversible,  the  p  v  diagram  would  be  a  continuation 
of  the  dotted  isothermal  D  E.  Let  the  expansion 
be  adiabatic  and  reversible,  the  cylinder  giving  up 
no  heat  to  the  steam,  the  curve  from  E  is  p  v?  = 
K  E  X  (M  K)7.  During  this  change  the  state 


point  of  the  0  <£  diagram  goes  down  vertically,  the 
entropy  of  the  steam  remaining  constant.  If  the 
expansion  were  reversible  and  the  cylinder  supplied 
some  heat,  but  not  enough  to  keep  the  steam  at 
1,000°  F.A.,  the  p  v  line  would  go  between  the 
isothermal  and  adiabatic,  but  for  simplicity  this 
part  will  be  taken  as  both  reversible  and  adiabatic, 


00  DIAGRAM   FOR   STEAM.          77 

and  therefore  isen tropic.  From  F  the  steam  will 
be  supposed  to  be  expanded  in  another  cylinder, 
which  abstracts  heat  during  the  expansion ;  so  that 
the  p  v  curve  F  H  bends  down  below  the  reversible 
adiabatic  line ;  and  the  state  point  on  the  6  </>  dia- 
gram moves  from  F  to  H,  showing  fall  of  tempera- 
ture and  decrease  of  entropy  going  on  together. 
This  curve  F  H  will  cut  the  saturation  line  at  G, 
so  that  at  H  about  0.2  of  the  steam  has  been  con- 
densed. From  H  to  I  let  the  expansion  be  neither 
adiabatic  nor  isothermal,  so  that  the  cylinder  walls 
give  up  heat  to  the  steam,  and  the  entropy  thus 
increases,  and  the  temperature  falls.  At  I  let  the 
exhaust  open  to  a  condenser  at  lower  temperature 
than  the  steam,  say,  as  represented  by  the  height 
of  the  line  M  J.  When  the  steam  has  all  fallen  to 
this  temperature,  the  entropy  will  be  reduced,  and 
the  state  point  will  be  at  the  point  L.  How  it  gets 
from  I  to  L  will  be  discussed  presently.  At  L  we 
have  nearly  a  pound  of  water  and  a  little  steam 
occupying  a  large  volume.  As  the  piston  comes 
back  the  steam  is  all  condensed  to  water,  and  we  get, 
not  to  A,  the  freezing-point,  but  to  M,  which  is 
not  at  zero  entropy. 

It  need  hardly  be  said  that  the  p  v  and  6  <f>  dia- 
grams here  discussed  are  not  practical  at  all;  they 
are  merely  put  together  on  paper  to  illustrate  the 
theory  of  entropy.  In  practice  the  indicator  diagram 
and  the  steam  per  half  stroke  is  the  starting  point, 


78 


THE    00   DIAGEAM. 


and  there  is  only  a  very  rough  approximation  to 
knowledge  of  what  happens .  In  the  present  examples 
the  0  <f>  diagram  is  made  up  from  steam  data  and  an 
imaginary  p  v  diagram,  as  if  of  a  single  cylinder 
with  no  clearance  or  cushioning,  and  many  pecu- 


liarities not  met  with  in  practice.  A  0  0  diagram 
with  straight  lines  and  corners,  and  a  right-hand 
side  like  Fig.  8,  is  not  found  in  steam  work. 

The  6  %  diagrams  need  not  be  discussed  further, 
as  it  is  beyond  the  scope  of  this  book  to  discuss 
factors  of  heat ;  they  were  introduced  here  to  show 
the  difference  between  the  6  <f>  and  real  heat  diagrams. 


6$  DIAGRAM   FOE   STEAM.  79 

In  order  to  keep  the  diagram  clear,  vertical  lines 
are  drawn  dotted  from  the  various  state  points  to 
the  base,  and  the  corresponding  letters  with  accents 
are  put  on  the  base  line  for  reference.  As  it  is 
tedious  to  follow  a  diagram  with  polygonal  areas 
denoted  by  numerous  corner  letters,  areas  are 
marked  with  small  letters,  each  letter  denoting  the 
area  enclosed  by  hard  lines.  Thus  a  +  b  is  the  area 
MBCDEFHIJLM. 

At  E  the  entropy  is  at  the  maximum  value 
touched  during  the  cycle.  To  find  the  incurred 
waste  at  E  the  increase  of  entropy  is  multiplied  by 
the  lowest  available  temperature.  For  simplicity 
the  same  pound  of  water  may  be  supposed  to  be 
used  over  and  over  again.  So  the  waste  is  not  the 
distance  P  E  multiplied  by  the  lowest  available 
temperature,  as  the  entropy  never  falls  below  E  M, 
corresponding  to  water  at  the  temperature  of  the 
condenser. 

Making  P  Q  =  R  M,  Q  E  is  the  maximum  increase 
of  entropy  during  the  cycle.  Choosing  32°  Fahr., 
or  493°  F.  A.  as  zero  of  entrop}T,  is  purely  arbitrary ; 
in  this  case  water  at  temperature  O  E  might  have 
been  taken  as  the  standard — that  is,  as  having  zero 
entropy.  Taking  Q  E  as  the  increase  of  entropy  at 
E,  the  next  question  is,  What  is  the  lowest  avail- 
able temperature  ?  The  lowest  temperature  reached 
is  at  L  M,  so  the  waste  incurred  up  to  the  point  E  is 
the  area  e  +  d ;  but  the  waste  is  really  considerably 


80 


THE    00   DIAGKAM. 


greater  owing  to  the  subsequent  treatment,  as 
the  temperature  of  the  condenser  is  not  really 
available,  except  for  the  small  outgiving  of  waste 
heat  between  L  and  M.  At  E,  however,  the  waste 
so  far  incurred  is  the  area  d  +  e.  A  perfect  engine 


working  from  E  by  reversible  changes  to  M,  without 
taking  in  any  more  heat,  would  move  the  state 
point  vertically  to  S,  and  then  across  to  M. 

The  path  from  E  to  F  calls  for  no  remark,  as  it 
is  assumed  to  be  reversible.  It  may  be  as  well  to 
point  out,  however,  that  a  vertical  line  on  the  0  <f> 
diagram  does  not  necessarily  mean  reversible  isen- 


6$  DIAGRAM   FOE    STEAM.  81 

tropic  adiabatic  change.  If  the  change  is  reversible 
and  isentropic,  it  is  also  adiabatic,  and  the  path  is 
vertical;  but  suppose  the  steam  were  wire-drawn 
between  E  and  F,  its  entropy  from  that  cause 
would  increase ;  but  if,  at  the  same  time,  it  lost 
heat  to  the  valve  or  other  metal-work  so  that  its 
entropy  was  reduced  by  loss  of  heat  exactly  enough 
to  counterbalance  the  increase  by  irreversible  expan- 
sion, the  trace  would  be  vertical.  These  doubly 
irreversible  disturbing  causes  would  have  additive 
effects  in  reducing  the  area  of  the  p  v  diagram 
relatively  to  the  0  <f>. 

Matters  are  complicated  in  Fig.  8  by  the  left- 
ward slope  F  H,  and  the  next  slope  H  I.  It  need 
hardly  be  repeated  that  such  a  corner  as  that  at  H 
does  not  occur;  it  is  shown  simply  as  an  example. 
At  E  the  incurred  waste  was  d  +  e.  From  F  to  H 
the  entropy  is  reduced,  and  at  H  the  incurred 
waste  is  M  T  H1  M1.  It  might  therefore  be  sup- 
posed that  from  F  to  H  the  actual  waste  is  the 
difference  in  entropy  of  F  and  H  multiplied  by 
M1  M.  But  M1  M  is  not  the  lowest  available  tem- 
perature here  at  all.  The  entropy  from  F  to  H 
has  been  reduced  by  giving  up  heat  at  a  varying 
higher  temperature,  not  to  a  condenser,  or  to 
anything  where  it  can  be  utilised,  but  to  cylinder 
walls,  where  it  may  be  conducted  away  and  wasted 
absolutely.  The  waste  corresponding  to  the  change 
F  H  is  thus  the  area  H  F  E1  H1.  If  the  heat  given 

£.  G 


82 


THE    6$  DIAGRAM. 


to  the  cylinder  walls  were  used  in  a  little  engine, 
some  of  this  would  be  recovered,  and  the  waste 
would  be  less.  The  waste  may  also  be  reduced  by 
some  of  the  heat  being  given  back  to  the  steam 
later  on. 


PTTC' 


From  H  to  I  the  steam  is  receiving  heat  from 
the  cylinder  walls.  If  this  heat  is  supplied  from 
the  furnace,  for  instance,  by  a  jacket,  the  incurred 
waste  at  I  exceeds  that  at  H  by  the  area  T  J  I1  H1 ; 
we  thus  have  a  vertical  strip  of  waste  area 
T  J  I1  H1  which  has  to  be  reckoned  twice.  If  the 
heat  supplied  between  H  and  I  was  some  of  that 


6$  DIAGRAM   FOR   STEAM.  83 

parted  with  between  F  and  H  to  the  cylinder  walls, 
the  waste  due  to  F  H,  which  was  taken  as  H  F  E1  H1, 
has  been  over-estimated,  and  instead  of  the  narrow 
strip  T  J  I1  H1  being  reckoned  twice,  it  must  be 
reckoned  once  only.  The  result  is  that  the  waste 
from  F  to  H — that  is,  the  heat  now  useless — is  the 
area  c  +  d,  so  that  the  strip  under  H  I  is  only 
counted  once,  and  the  little  area  H  I  J  T  would  be 
saved,  or  utilised  if  the  lower  line  M  J  of  the  closed 
cycle  were  the  lowest  available  temperature.  If  the 
curve  I  K  L  is  the  lowest  available  temperature,  the 
area  HUT  is  reduced  to  a  little  triangular  piece 
at  the  top. 

The  next  point  is  the  passage  from  I  to  L.  The 
difficulty  is  that  while  the  steam  is  blowing  through 
the  exhaust,  the  part  that  is  condensed  is  at  con- 
denser temperature,  while  the  part  that  is  in  the 
cylinder  begins  by  being  at  a  higher  temperature, 
and  the  temperature  gradually  falls.  There  is  no 
one  value  for  the  temperature  during  the  change. 
The  best  thing  to  do,  therefore,  is  to  make  some 
assumption  which  will  cause  no  error  in  deriving 
results  from  the  0<f>  and  p  v  diagrams,  and  will  at 
the  same  time  give  a  meaning  to  0  during  this 
change.  One  method  is  to  assume  that  the  steam, 
instead  of  rushing  off  to  a  condenser,  is  cooled  to 
condenser  temperature  at  constant  volume.  This 
gives  the  curve  I  K  L  (Fig.  8).  Another  assump- 
tion might  be  that  the  steam  expands  adiabatically 


84 


THE    e<l>  DIAGRAM. 


and  reversibly  to  the  condenser  temperature,  and  is 
then  compressed  isothermally  to  L.  This  involves 
external  work,  and  is  therefore  incorrect.  The 
condensation  at  constant  volume  is  correct,  as  it 
gives  the  right  result.  By  opening  the  exhaust  at 


I  the  lowest  available  temperature  is  raised  to  I, 
and  falls  along  the  path  I  K  L.  The  area  I  K  L  J 
is  thus  added  to  the  actual  waste. 

The  meaning  of  this  slice  off  the  0  <£  diagram  is 
not  difficult  to  catch.  As  has  been  explained 
already,  the  statement  generally  made  that  the  area 
of  the  0  <£  cycle  is  the  same  as  that  of  the  indicator 


6$  DIAGRAM   FOR   STEAM.  85 

diagram  is  wrong.  It  is  always  greater,  and  the 
difference  is  the  ungained  work  and  the  "lost 
work,"  which  has  not  yet  been  discussed.  Cutting 
off  a  corner  like  I  K  L,  therefore,  looks  like  re- 
ducing the  area  of  the  6  <£  cycle,  so  as  to  lessen  the 
difference  of  area,  and  therefore  lessen  the  ungained 
work.  It  might  therefore  appear  that  the  0<f> 
diagram  showed  that  exhausting  above  condenser 
temperature  is  rather  a  good  thing  than  otherwise, 
and  therefore  the  6<i>  diagram  is  misleading.  This 
would  be,  however,  an  incorrect  view.  The  ungained 
work  is  not  lessened  by  raising  the  lower  side  of  the 
0<£  cycle.  Thus,  if  there  was  some  ungained  work 
during  the  increase  of  entropy  from  B  to  C,  it 
might  be  regarded  as  represented  by  an  area  of 
height  B  B  or  C  C,  and  width  equal  to  the  uncom- 
pensated  entropy.  It  stretches  right  down  to  the 
bottom  of  the  diagram.  This  is  an  area  by  which 
the  6  (f>  cycle  exceeds  the  p  v  cycle,  therefore,  what- 
ever the  height  of  the  lower  line  of  the  0$  cycle. 
If  the  lower  side  of  the  #<£  cycle  were  raised  until 
it  coincided  with  the  upper,  and  the  ungained  work 
were  as  before,  the  area  of  the  p  v  cycle  would  be 
negative,  and  numerically  equal  to  the  ungained 
work  ;  that  is  to  say,  the  steam  would  expand  doing 
no  external  work,  and  the  engine  would  do  work 
equal  to  the  ungained  work  in  compressing  it  into 
water  again. 

The   theory   of  ungained   work   may   be  shown 


THE   04,   DIAGKAM. 


better  by  a  simple  diagram.  Suppose  water  at  A 
in  the  0<f>  and  p  v  diagrams,  Fig.  11,  were  heated 
reversibly  to  B,  then  expanded  to  C  reversibly  and 
isothermally,  and  then  expanded  to  D  reversibly 
and  adiabatically,  and  finally  compressed  to  A 
again,  also  reversibly.  Assume  the  p  r  diagram, 
Fig.  11,  to  be  drawn  to  correspond  in  scale  with 
the  0  <£,  and  let  the  areas  be  cut  up  by  the  vertical 
and  horizontal  lines  into  small  areas  which  do  not 
overlap,  each  with  its  letter  of  reference.  Taking 


the  four  steps,  and  putting  down  the  intake  of  heat 
and  output  of  work,  we  have  : — 


Step. 
A  to  B 
B  „  C 
C  „  D 
D  „  A 
Balance 

Heat  Taken  in. 
a  -f  d 

1)  +  e  +  c  +f 
0 
-(/+<?  -M) 
a  -f-  b  -f  c      a 

Work  Done. 
0 

c+f=fj+j 
a  4-  ft  =  k  +  i 
—  /=  —  (i  - 

Heat  Taken  in. 

Work  Done. 

a  +  d 

0 

b+  e 

0 

0 

a  4.  b  =  h  +  i 

—  ( 

f+e+d) 

-/  =  -  (i+; 

a  +&-/        a+i-f=h-j 

6<j>  DIAGRAM   FOR   STEAM.  87 

If,  however,  the  expansion  were  wholly  irre- 
versible, to  take  an  extreme  case  for  simplicity,  the 
statement  would  be  : — 

Step. 
A  to  B 
B  „  C 
C  „  D 
D  „  A     - 


The  area  of  the  0  <£  cycle  would  be,  as  before, 
a  +  b  +  c,  but  thepv  cycle  would  be  smaller  by  the  strip 
c  -f-  /.  If  the  lowest  available  temperature  is  raised, 
it  does  not  reduce  this  difference,  as  it  is  on  both  sides 
of  the  line  of  the  lowest  available  temperature.* 

Turning  back  to  Fig.  8,  cutting  off  the  area  by 
the  curve  I K  L  does  not  make  the  efficiency  higher, 
according  to  the  6$  diagram,  though  the  waste  is 
the  difference  of  areas  of  the  p  v  and  6  <£  cycles,  and 
though  cutting  off  this  corner  lessens  the  6  <£  cycle. 
If  the  path  had  been  by  I J  L,  the  steam  would  have 
been  expanded  further,  and  more  work  would  have 
been  obtained.  If  the  expansion  from  I  to  J  and  the 
compression  from  J  to  L  were  reversible,  both  p  v 
and  6<j>  cycles  would  be  equally  enlarged  by  the 

*  For  simplicity  it  is  here  assumed  that  at  the  lower  tem- 
perature U  =  d  -f-  e ;  it  is  in  fact  a  little  less.  This  does  not 
reduce  the  area  c  -f  /,  so  the  proof  is  not  affected  by  the 
slight  inaccuracy  of  the  assumption. 


88 


THE 


DIAGRAM. 


course  taken.  In  cutting  off  the  corner  what  is 
really  done  is  to  increase  the  waste  \>y  raising  the 
lowest  available  temperature.  This  is  another  form 
of  the  same  thing  as  occurred  from  G  to  H.  Thus 
at  I  the  waste  so  far  incurred  is  e,  part  of  the  area 


N  I  J  M  being  still  to  be  had  as  work ;  but  if  the 
exigencies  of  the  case  call  for  a  raising  of  the  lowest 
available  temperature,  there  is  a  reduction  of  the 
efficiency  of  the  engine,  and  an  increase  of  the  waste, 
but  not  an  increase  of  difference  of  cycle  areas.  The 
corner  area  I  J  L  K  is  not  ungained  work  in  the 


00  DIAGKAM   FOR   STEAM.  89 

sense  the  term  has  been  employed  (fo  d(f>  —  h), 
due  to  irreversibility ;  it  is  increased  waste  due  to 
raising  the  lower  temperature  limit. 

The  irregular  area  b  +  c  +  d  +  e  is  the  total 
reversible  waste,  assuming  that  the  entropy  from  H 
to  I  was  increased  by  some  of  the  heat  given  out 
between  F  and  H.  It  must  not  be  supposed,  how- 
ever, that  the  waste  is  necessarily  no  greater  than 
this. 

So  far  irreversible  compression  has  not  been  dis- 
cussed. If  a  cylinder  of  gas  or  steam  is  compressed 
by  using  up  external  work,  it  is  difficult  to  think  of 
any  form  of  irreversible  compression  which  would 
use  up  more  work  and  give  out  more  heat  than 
would  account  for  the  decrease  of  the  steam's 
entropy.  In  a  turbine  acting  as  a  pump,  however, 
the  waste  would  be  considerably  greater  than  the 
area  under  the  6<f>  cycle.  This  is  pointed  out  to 
prevent  any  such  mistake  as  supposing  the  area 
under  the  0<t>  cycle  to  be  the  only  waste.  The 
waste  must  be  at  least  that  area,  and  it  may  be 
more.  This  increased  waste  appears,  however,  as 
a  reduction  of  the  p  v  cycle  area,  as  the  increased 
waste  due  to  irreversible  compression  increases  the 
work  put  into  the  working  substance  by  raising  the 
lower  side  of  the  p  v  diagram's  cycle.  It  thus  shows 
the  difference  of  area  of  the  6  <£  and  the  p  v  cycles. 
The  excess  of  area  of  the  6  <£  over  the  p  v  cycle  is 
thus  made  up  of  two  parts  :  the  ungained  work,  due 


90  THE    6$  DIAGEAM. 

to  irreversible  expansion,  and  lost  work,  due  to 
irreversible  compression. 

In  practical  engineering,  in  the  case  of  a  recipro- 
cating engine,  we  have  some  approach  to  the  p  v 
diagram  given  by  the  indicator ;  but  there  is  un- 
certainty as  to  the  value  of  v  at  the  beginning  of 
the  stroke — that  is,  until  the  admission  is  finished. 
Again,  there  is  uncertainty  if  the  exhaust  opens 
gradually.  Moreover,  the  steam  may  not  be  at  the 
same  temperature  throughout  its  volume  at  any 
given  time.  An  approximately  true  p  r  diagram  is 
made,  and  the  0$  inferred  from  it  by  means  of 
tables  or  Sankey's  chart.  The  difference  of  areas 
of  the  cycles  should  then  be  measured,  to  give  the 
ungained  work,  and  the  wasted  work,  and  the  lower 
boundary  of  the  cycle  should  be  examined  to  see 
whether  the  waste  can  be  reduced  anywhere  by 
keeping  down  the  lower  temperature  limit. 

Ungained  work  due  to  irreversible  expansion  in 
the  cylinder  is  too  minute  to  matter  in  a  recipro- 
cating engine,  but  in  the  turbine  it  may  be  a  very 
serious  matter.  The  steam  turbine  may  in  the 
future  be  very  much  more  carefully  studied  thermo- 
dynamically  than  the  reciprocating  engine.  But 
there  is  a  good  deal  of  ungained  work  in  an  ordinary 
engine  in  wire-drawing  through  valves  and  ports. 

The  0<f>  cannot  well  be  compared  with  the 
indicator  diagram  in  the  case  of  a  turbine.  It  is  in 
fact  somewhat  difficult  to  find  out  what  is  going  on 


6$  DIAGRAM   FOR   STEAM.  91 

inside  the  case.  If  it  is  driving  a  dynamo  the  output 
and  the  steam  supply  are  known  and  the  pressure, 
temperature,  and  dryness,  of  the  steam  as  it  enters. 
The  temperature  can  be  taken  by  inserting  small 
thermometers  or  thermo-couples  at  various  points. 
The  velocity  may  be  not  great  enough  to  prevent 
the  pressures  being  taken  from  point  to  point ;  and 
perhaps  some  conclusions  as  to  the  dryness  at 
various  vanes  can  be  obtained. 

If  the  condition  of  the  steam  entering  is  known, 
and  the  temperature  of  the  condenser,  the  maximum 
or  ideal  efficiency  is  known,  and  an  ideal  6<j>  diagram 
can  be  plotted.  If  the  turbine  is  jacketed  the  jacket 
steam  must  be  included.  From  the  pressure  tem- 
perature and  wetness  of  the  exhaust  steam,  its 
specific  entropy  can  be  found  and  marked  on  the 
0<J>  curve,  and  an  equivalent  exhaust  curve  drawn, 
as  in  Fig.  8.  If  the  turbine,  whether  jacketed  or 
not,  does  not  lose  heat  perceptibty  by  radiation  and 
convection,  the  state  point  for  the  exhaust  will  be 
to  the  right  of  that  for  admission,  and  lower  in 
temperature  of  course.  From  the  diagram  so 
obtained  the  "  ungained  work "  can  be  found. 
The  inefficiency  will  be  the  result  of  irreversible 
expansion,  and  conduction  of  heat  along  the  metal- 
work  of  the  turbine.  This  tells  the  result  of  the 
whole  turbine,  and  when  several  turbines  are  used 
in  series  that  is  specially  valuable,  corresponding 
roughly  to  the  information  as  to  each  of  the 


92  THE    00   DIAGRAM. 

cylinders  in  a  reciprocating  engine.  If  the  necessary 
information  can  be  obtained  from  point  to  point 
throughout  the  turbine,  the  localisation  of  blame  is 
possible,  and  the  designer  can  find  out  where  the 
steam  is  rushing  through  too  easily  without  doing 
its  proper  work. 

Into  the  details  of  the  application  of  the  0<f> 
diagram  to  actual  practice  it  is  not  my  object  to 
go.  That  may  be  left  for  a  future  opportunity,  or 
the  work  may  be  done  better  by  others.  But  the 
first  thing  is  to  get  the  theory  of  the  9  <£  diagram 
cleared  of  the  common  errors  about  it,  so  that  its 
real  meanings  can  be  brought  out. 

The  present  position  of  the  0  <j>  diagram  in  engi- 
neering is  that  it  is  discussed  in  college  text-books 
and  in  treatises  for  engineers ;  and  it  is  said  to  be 
very  instructive,  and  to  give  great  insight  into  the 
economy  of  the  steam-engine.  But  it  has  made 
little  or  no  progress  in  real  engineering.  One 
engineer,  of  high  rank,  has  been  said  to  have  used 
it,  and  got  benefit  from  it.  But  so  long  as  it  is 
supposed  that  entropy  isj  dh/0,  or  heat-weight,  or 
a  factor  of  heat,  or  that  the  6$  is  a  heat  diagram, 
or  that  its  area  is  in  British  thermal  units,  or  that 
the  0  <£  diagram  is  of  the  same  area  as  the  p  v  cycle, 
it  seems  likely  to  remain  practically  useless.  It 
will  be  treated  with  great  respect,  because  people 
always  respect  what  they  do  not  understand,  espe- 
cially if  it  looks  at  all  mathematical.  Engineers 


w^-v 
OF  THE 

UNIVERSITY 

OF 


DIAGRAM   FOR   STKfflrT^       93 


will  not  like  to  admit  that  they  do  not  understand 
it;  or,  at  most,  those  who  have  neglected  mathe- 
matics will  regret  that  the  calculus  was  not  taught 
apprentices  in  their  young  days,  and  recommend 
young  men  to  take  the  0  <f>  diagram  to  their  bosoms 
and  thank  their  stars  for  it.  They  may  even  tackle 
one  of  the  numerous  and  excellent  books  of  potted 
mathematics  with  which  the  benighted  engineer  is 
now  befriended.  If  they  can  digest  the  potted 
mathematics  with  any  sort  of  comfort,  they  are 
certain  to  be  saddened  internally  by  the  0  <{>  diagram 
as  at  present  constituted.  But  they  feel  it  is  a 
wonderful  thing,  which  ought  to  be  swallowed. 


CHAPTER  IV. 

CONDUCTION. 

Movement  of  Entropy. — When  one  body  is  in 
contact  with  another  and  loses  heat  to  it,  it  is  usual 
to  say  that  the  heat  moves  from  the  hotter  to  the 
colder  body,  just  as  if  the  heat  were  caloric,  that 
moved  from  one  place  to  another.  But  what  really 
happens  is  that  the  heat  of  one  body  increases  as 
the  other  diminishes.  Similarly,  energy  is  said  to 
move.  Light,  electricity,  sound  and  waves  on  water 
are  all  regarded  as  moving.  When  a  body  is  in 
contact  with  another,  and  decreases  its  entropy  at 
the  same  rate  as  the  other  increases,  it  is  most  con- 
venient to  think  of  entropy  as  moving  from  one  to 
the  other,  just  as  we  think  of  heat  moving.  As  the 
entropy  of  a  body  cannot  decrease  without  heat 
coming  out,  while  it  can  be  increased  without  heat 
coming  in,  it  is  easy  to  think  of  entropy  as  moving 
from  one  body  to  another,  and  as  increasing  or 
growing  in  any  body,  but  never  as  disappearing. 
Thus  entropy  may  move,  and  may  grow,  but  never 
disappears.  There  are  sources,  but  no  sinks.  I 
do  not  know  if  entropy  is  considered  as  moving 
in  any  treatises  on  thermodynamics,  but  think 


CONDUCTION   OF   HEAT.  95 

not ;  but  that  does  not  make  the  idea  any  less 
useful  or  valuable. 

Conduction  of  Heat. — Conduction  of  heat  is  a 
very  important  irreversible  process.  Suppose  a  hot 
body,  say  a  vessel  containing  10  Ibs.  of  water  at 
650°  F.A.,  its  entropy  being  2.79,  is  put  in  contact 
with  another  of  the  same  size,  but  at  a  temperature 
of  550°  F.A.,  its  entropy  being  1.09,  the  two  will 
come  to  the  mean  temperature  of  600°  F.A.,  and 
their  entropies  will  be  1.97  each,  or  3.94  together. 

The  total  entropy  has  thus  gone  up  from  3.88  to 
3.94,  showing  incurred  waste  of  0,06  multiplied  by 
the  lowest  available  temperature.  Suppose  the 
lowest  available  temperature  to  be  550°,  the  waste 
is  thus  33  British  thermal  units. 

If  the  10  Ib.  of  water  at  650°  were  put  in 
connection  with  a  thermodynamic  engine  which 
could  take  heat  at  a  gradually  falling  temperature 
and  return  the  waste  heat  at  550°,  a  certain  output 
in  work  would  be  obtained.  Of  course  the  water 
cannot  give  out  all  its  heat  at  650°;  the  mutivity 
(#i  ~~  0a)'0i>  or  changeability  of  the  heat  into  work,  falls 
with  the  temperature.  The  water  loses  as  much  heat 
between  650°  and  600°  as  between  600°  and  550°, 
but  more  of  the  heat  given  out  between  650°  and 
600°  is  convertible  into  work.  If  one  vessel  gives 
heat  to  a  thermodynamic  engine  from  650°  down  to 
550°,  there  will  thus  be  more  work  obtainable  than 
if  two  gave  out  heat  from  600°  to  550°. 


96 


CONDUCTION. 


This  is  shown  clearly  by  the  0  <£  diagram.  Fig.  12 
is  the  0$  diagram  for  10  Ib.  of  water  from  550°  to 
650°  F.  A.  The  area  A  C  F  I  represents  Ui  -  U2,  or 
U  at  650°  F.A.,  taking  550°  as  zero.  It  is  obviously 
equal  to  10  X  100,  or  1,000  British  thermal  units. 
The  available  part  of  the  energy  is 

W= 


The  first  term  is  the  whole  of  the  heat  needed 
to  raise  the  temperature  from  550°  to 
650°,  or  1,000  British  thermal  units, 
and  is  shown  by  the  area  A  C  F  I. 
The  second  term  is  the  waste  A  D  F  I, 
for  A  D  =  $  =  10  log.  OjOto  and  03  is 
I  A.  So  the  available  part  is  the  area 
A  C  D.  If  a  vertical  line  B  G  is 
drawn  so  that  G  B  =  600°,  A  E  is 
the  entropy  of  10  Ib.  of  water  at  that 
temperature,  taking  550°  as  zero ;  and 
A  B  G  I  is  500  British  thermal  units. 
One  vessel  of  water  thus  has  the  heat 

I          Q 

A  C  D  available  or  convertible  into 
work  to  start  with,  while  the  other  has  none.  On 
equalising  temperatures  there  are  two  vessels,  each 
with  only  ABE  available  ;  and  A  C  I)  is  obviously 
greater  than  twice  ABE.  As  the  area  A  B  G I  is 
equal  to  B  C  F  G,  A  E  is  greater  than  E  D,  so  the 
entropy  A  D  of  the  vessel  at  650°  is  less  than 
that  of  the  two  vessels  at  600°,  or  twice  A  E, 


CONDUCTION   OF  HEAT.  97 

and  the  waste,  twice  A  E  G   I,  is  greater  than 
AD  FI. 

When  a  hot  body  communicates  heat  to  one  of  a 
lower  temperature,  there  is  thus  incurred  waste  or 
increase  of  entropy  of  the  two.  Earlier  in  this  book 
reversible  changes  have  been  discussed  in  which 
heat  went  from  a  reservoir  to  a  perfect  gas  and 
expanded  it,  and  so  on.  The  provision  was  made, 
however,  that  the  difference  of  temperature  was 
sensibly  nothing,  so  as  to  give  no  perceptible  down 
grade  of  temperature,  and  no  increase  of  entropy 
due  to  irreversible  conduction. 

It  has  been  shown  that  if  a  hot  and  cold  body 
are  put  in  contact  with  each  other  and  allowed  to 
equalise  their  temperature,  there  is  a  growth  of  en- 
tropy ;  but  only  the  final  result  has  been  considered. 
It  may  be  as  well  to  discuss  what  happens  during 
the  change.  Suppose  a  reservoir  of  heat — that  is 
to  say,  an  ideal  bod}7  so  large  that  it  can  give  out 
heat  without  falling  in  temperature — is  put  in  con- 
tact with,  say,  a  mass  of  iron  at  a  lower  temperature. 
Suppose  for  the  moment  that  the  conductivity  of 
the  ideal  body  is  so  enormous  that  the  part  of  it 
against  the  iron  remains  at  the  same  temperature 
as  the  rest  of  the  reservoir,  say  Olt  The  face  of  the 
part  of  the  iron  block  touching  the  reservoir  is  then 
at  temperature  Olm  If  the  block,  to  begin  with,  is 
all  at  uniform  temperature,  as  soon  as  contact  takes 
place  the  block  is  at  #1  on  one  face,  and  at  lower 

E.  H 


98  CONDUCTION. 

temperatures  elsewhere,  and  the  temperature  of  the 
block  is  no  longer  uniform.  In  a  given  time  the 
reservoir  has  lost  heat,  say  HI,  at  constant  tempera- 
ture QI,  so  its  entropy  has  been  reduced  by  HI/#I. 
The  block  has  received  heat  HI  at  6\,  as  #1  is  the 
temperature  of  the  face  or  part  of  the  envelope 
through  which  the  heat  HI  passes,  so  its  entropy 
increase,  as  far  as  heat  from  outside  is  concerned, 
is  HI/#I.  But  at  the  end  of  the  process,  when  the 
block  has  come  to  temperature  Olt  and  taken  in  HI, 
its  increase  of  entropy  is  greater  than  HI/#I.  There 
has  been  a  growth  of  entropy  in  the  volume  of  the 
block.  This  volume  growth  is  evidently  the  un- 
compensated  entropy  due  to  the  irreversible  nature 
of  conduction.  If  the  block  takes  in  heat  HI  at 
temperature  0i,  its  increase  of  entropy  is  thus  greater 
than  HI/#I  ;  and  if  the  temperature  of  the  face  varies 
during  the  change,  the  entropy  taken  in  isy  d  H/0 — 
the  compensated  entropy ;  but  the  entropy  of  the 
block  is  greater  ihaufd  H/0  by  the  uncompensated 
entropy,  or  $  >J  d  H/0,  as  in  all  irreversible  or 
real  changes.  It  must  be  borne  in  mind  that  0  is 
the  temperature  of  the  face  of  the  block  in  contact 
with  the  reservoir,  not  the  temperature  of  the  block. 
The  temperature  of  the  block  varies  from  part  to 
part.  To  find  how  the  entropy  grows  during  con- 
duction, imagine  a  bar  of,  say,  copper  of  uniform 
section,  say,  a  square  inch.  Let  it  be  kept  hot  at 
one  end  and  cold  at  the  other,  and  be  wrapped  in 


CONDUCTION   OF   HEAT.  99 

heat-insulating  lagging  along  its  length.  Imagine 
a  surface  cutting  this  bar  across  so  that  all  this 
surface  is  at  one  temperature  0i.  The  copper  on 
each  side  of  this  surface  is  either  hotter  or  colder, 
but  the  surface  is  at  01}  and  remains  at  0i9  because 
the  copper  is  in  a  steady  state,  with  heat  flowing 
along  steadily  and  continually  at  uniform  rate.  An 
inch  along  toward  the  colder  end  imagine  a  second 
surface  to  cut  the  bar  at  right  angles,  and  let  its 
temperature  be  0*  Let  Ol  be  132°  Fahr.,  and  02  be 
32°  Fahr.  Then  the  flow  of  energy,  calculated 
*rom  the  thermal  conductivity  of  copper,  is  about 
0.5  British  thermal  unit  per  second.  Through  the 
hotter  face  we  thus  have  each  second  H  =  0.5  and 
#1  =  593°  F.A.,  so  there  is  an  intake  of  entropy  of 
0.00083  per  second.  At  the  cold  face  there  is  an 
outflow  of  H  =  0.5  at  02  =  493°,  and  the  outflow  of 
entropy  per  second  0.00103.  There  is  thus  a 
volume  growth  of  entropy  of  0.0092  per  second  in 
the  cubic  inch  of  copper.  If  freezing-point  is  the 
lowest  temperature,  the  waste  due  to  the  irreversi- 
bility  comes  out  at  0.14  British  horse-power. 

Whenever  there  is  conduction  of  heat  there  is 
thus  growth  of  entropy  throughout  the  volume  of 
the  conducting  body.  Thus  the  fall  of  temperature 
between  the  immediate  surface  of  the  hot  coal  and 
the  water  is  all  by  continuous  temperature  gradients 
• — that  is  to  say,  by  conduction. 

There  is  great  growth  of  entropy  in  the  furnace 

ii  2 


100  CONDUCTION. 

gas  between  the  spot  where  the  conduction  takes 
place,  and  the  boiler  flues  or  tubes.  There  is 
growth  of  entropy  in  the  external  scale  or  dirt ;  a 
little  growth  in  the  metal  separating  the  gases  from 
the  water ;  considerable  growth  in  the  scale.  Boiler 
scale  is  a  sort  of  entropy  hot-bed.  There  is  more 
growth  of  entropy  in  conduction  in  the  boiler ;  but 
this  is  not  serious.  Then  there  is  growth  in  the 
metal  of  the  valve  that  separates  steam  at  different 
temperatures.  The  growth  of  incurred  waste  in  a 
large  slide  valve,  such  as  used  to  be  employed  in 
marine  engines,  may  amount  to  about  10  British 
horse-power,  though  there  is  no  loss  of  heat.  Then 
there  is  waste  in  conduction  from  the  jacket  to  the 
cylinder,  from  the  steam  to  the  metal  walls  and 
back  again,  and  so  on.  All  this  growth  of  entropy 
means  waste  of  energy  that  it  is  the  engineer's 
object  to  keep  down.  All  this  entropy  has  nothing 
to  do  with  the  waste  of  heat  by  radiation  and  con- 
duction. When  heat  is  radiated  from  a  pipe,  for 
example,  at  the  rate  of,  say,  enough  British  thermal 
units  per  second  to  amount  to  10  British  horse- 
power, whatever  that  number  may  be,  it  does  not 
deduct  10  British  horse-power  from  the  engine 
output,  because  the  engine  would  only  have  used 
some  of  the  heat  anyhow. 

The  growth  of  entropy  in  conduction  of  heat  is 
not  generally  treated  in  this  way ;  in  fact,  the  method 
may  be  original.  Writers  on  thermodynamics — 


CONDUCTION    OF   HEAT.  101 

even  specialists — do  not  usually  give  any  indication 
of  what  they  mean  by  0  or  p  in  irreversible  changes. 
In  discussing  conduction  in  reversible  changes,  0  is 
the  temperature  of  the  reservoir  or  of  the  working 
substance,  or  both,  as  it  is  the  same,  and  the  entropy 
equation  isJdH/0  =  3>.  Then  in  the  case  of 
irreversible  change  we  are  frequently  told  that 
J  (H{/0  =  ^  is  still  true,  which  is  nonsense  anyhow, 
or  that  fd  H/0  <  3>,  which  is  vague  if  we  are  not 
told  what  6  is.  There  may  be  no  reservoir,  as  the 
heat  H  may  come  from  a  body  whose  temperature 
is  not  uniform.  If  the  working  substance  is  also  of 
varying  temperature,  0  cannot  mean  the  temperature 
of  the  working  substance.  Personally,  I  have 
always  read  6  to  be  the  temperature  of  the  bounding 
surface  through  which  the  heat  H  passes,  as  that 
is  the  only  reading  which  conveys  any  physical 
meaning  to  me ;  but  I  have  since  found  I  am 
rather  peculiar  in  this.  What  idea  is  conveyed  by 
J  d  H/0  <  $  to  those  who  do  not  think  of  0  as  the 
temperature  of  the  bounding  surface  I  cannot  con- 
ceive. There  seems  to  be  a  large  class  of  people 
who  are  mathematically  trained  only  to  the  point 
of  having  facility  in  the  blind  manipulation  of 
mathematical  symbols,  and  they  have  the  extra- 
ordinary faculty  of  being  able  to  read,  and  even 
write,  mathematical  symbols,  and  to  get  correct 
results  without  having  any  clear  idea  of  what  they 
are  doing.  Of  course,  this  involves  a  very  degraded 


102  CONDUCTION. 

type  of  mathematics.  Sometimes  it  is  held  that  0 
still  means  the  temperature  of  the  conducting  body, 
and  the  body  must  be  split  up  into  elements  each 
so  small  that  its  temperature  is  sensibly  constant. 
Thus,  if  there  are  two  bodies  at  different  tempera- 
tures, but  each  at  uniform  temperature  through- 
out, the  hot  one  may  give  heat  slowly  to  the  cold 
one.  When  they  are  equalised,  the  total  entropy  has 
increased.  Of  each  it  is  true  that  fd  H/#  =  <I>, 
where  0  is  the  temperature  of  the  body.  It  is, 
therefore,  it  is  said,  true  of  each  element  of  a  body, 
if  the  body  is  not  at  uniform  temperature  ;  but  the 
elements  are  so  small  that  each  is  at  a  uniform 
temperature,  so  that  for  the  whole  body  fd  H/0 
=  <£.  Of  course,  this  is  dreadful  confusion  ;  H  and 
6  have  different  meanings  in  the  two  cases.  The 
underlying  idea  is  also  confused.  If  a  hot  and  a 
cold  bod}7  are  each  at  a  uniform  temperature,  and 
the  hot  gives  up  heat  to  the  cold,  there  must  be 
something  between  them  in  which  the  conduction 
of  heat  and  temperature  gradient  takes  place.  That 
something,  in  a  given  time,  takes  in  H  at  Ol  and 
gives  out  H  at  0.2,  if  not  itself  absorbing  or  giving 
out  heat.  The  entropy  growth  is  therefore  going 
on  in  it.  The  growth  goes  on  when  there  is  con- 
duction of  heat  and  a  down  grade  of  temperature ; 
and  to  assume  that  a  conducting  body  can  be  divided 
up  into  elements,  each  uniform  in  temperature, 
involves  the  body  being  made  up  of  something  like 


THE    UNIT    OF   ENTROPY.  103 

the  bricks  of  a  house  with  hadly  conducting  mortar 
between  them.  The  mortar  may  be  thinned  with- 
out limit,  but  if  the  down  gradient  of  temperature 
is  there  and  a  flow  of  heat  across,  the  uncompen- 
sated  entropy  grows  entirely  in  the  mortar.  To 
neglect  the  mortar  is  to  make  out  that  a  body  con- 
ducting heat  is  made  up  of  elements,  each  of  which 
is  undergoing  a  reversible  change,  while  the  whole 
undergoes  an  irreversible  change.  The  argument 
is  like  saying  that  a  path  down  a  hill  can  be  divided 
up  into  elements  so  small  that  the  height  of  each  is 
uniform.  Water  running  over  each,  therefore, 
does  no  work,  so  water  running  down  the  hill  does 
no  work.  This  point  is  emphasised  here  because 
it  is  not  generally  realised  that  there  is  a  volume 
growth  of  uncompen sated  entropy  wherever  there 
is  a  temperature  gradient.  This  is  a  very  convenient 
idea,  as  it  shows  that  all  conduction  or  radiation  of 
heat  involves  waste,  the  power  wasted  depending 
solely  on  the  temperature  and  its  rate  of  decrease, 
and  the  conductivity  of  the  material. 

The  Unit  of  Entropy. — Nomenclature  is  one  of 
the  weak  points  of  thermodynamics.  There  is  no 
name  for  the  unit  of  temperature,  and  there  are  two 
thermometer  scales  used  in  England  and  two  on  the 
Continent.  None  reads  exactly  in  accordance  with 
the  Kelvin  or  perfect  gas  scale.  The  unit  of  heat, 
which  is  quite  an  unnecessary  nuisance,  has  no  name, 
for  British  thermal  unit  is  not  a  name  ;  it  is  an 


104  CONDUCTION. 

opprobrious  epithet.  The  unit  of  entropy  has  not 
even  that.  It  might  be  called  the  British  entropic 
unit.  I  have  elsewhere  used  the  term  "  entrop  "  to 
denote  the  unit,  also  the  term  "  claus,"  short  for 
Clausius,  for  the  practical  unit  of  entropy  on  the 
C.G.S.  system.  It  is  usual  to  derive  the  practical 
unit  on  the  C.G.S.  system  from  a  man's  name,  as 
the  Ohm,  Volt,  Joule,  &c.  The  claus  is  the  entropy 
which,  when  multiplied  by  the  lowest  available 
temperature  in  absolute  Centigrade  degrees,  gives 
the  incurred  waste  in  joules,  not  in  calories,  either 
of  the  big  kind  or  of  the  little  kind,  or  of  the  middle- 
sized,  if  there  is  one  yet.  The  British  entropic 
unit,  as  used  in  this  book,  is  the  entropy  which, 
when  multiplied  by  the  lowest  available  absolute 
temperature,  gives  the  incurred  waste  in  British 
thermal  units.  "  Entrop  "  is  shorter  and  more  con- 
venient than  "  British  entropic  unit."  It  does  not 
matter  which  thermometer  scale  is  used  in  the 
case  of  the  entrop,  for  if  a  body — say  10  Ib.  of 
water — at  the '  temperature  of  boiling  water  has 
3.14  entrops— the  waste,  if  all  the  180  British 
thermal  units  are  given  out  at  freezing  temperature, 
will  be  180  by  3.14  British  thermal  units.  But  if 
degrees  Centigrade  are  used,  there  will  only  be 
100  thermal  units,  whose  nationality  it  may  be 
difficult  to  get  an}'  civilised  country  to  acknowledge, 
so  the  waste  will  be  100  X  3.14,  which  is  the  same 
heat  as  before.  The  claus  refers  to  absolute  Centi- 


PHYSICAL   MEANING   OF' ENTEOPY.     105 

grade  degrees  only,  as  the  alteration  of  the  thermo- 
meter scale,  though  it  would  change  the  magnitude 
of  each  of  the  various  calories,  would  not  affect 
the  joule. 

It  may  seem  strange  that  a  standard  state  has  to 
be  taken  for  zero  entropy,  instead  of  simply  finding 
the  total  entropy  of  a  body  and  stating  it.  The 
usual  reason  given  is  that  the  total  entropy  of  a  body 
is  unknown.  The  real  reason  is  that  the  entropy  of 
a  body  is  necessarily  infinite.  » 

If  a  body  of  unit  mass  is  imagined  at  zero  tem- 
perature, and  that  is  taken  as  zero  entropy,  because 
as  no  heat  can  be  given  out  no  further  reduction  of 
entropy  is  possible ;  to  heat  it  up  to  any  finite 
temperature  0,  the  increase  of  entropy  necessary  is 
y^  d0/0  =  <£.  So  <£  =  log.  $i  -  log.  00 ;  but  log.  00  or 
log.  0  is  negative  infinity,  so  the  entropy  needed 
to  raise  the  bod}7  to  any  finite  temperature  is 
infinite. 

Physical  Meaning  of  Entropy. — So  far  entropy  has 
been  treated  as  a  function  of  the  co-ordinates,  or  of 
the  incurred  waste ;  but  the  question  arises  as  to 
whether  it  has  any  further  physical  meaning.  For 
instance,  if  a  body  is  heated,  we  picture  the  change 
of  temperature  as  being  accompanied  by  change  of 
molecular  motion.  The  difficulty  with  entropy  is 
that  it  is  a  quantity  whose  increase  denotes  incurred 
waste,  and  it  is  thus  very  general,  so  that  entropy  is 
increased  in  different  ways  in  different  cases,  and 


106  CONDUCTION. 

there  is  no  one  molecular  change  that  corresponds 
with  increase  of  entropy. 

Take  first  the  case  of  a  perfect  gas.  The  tem- 
perature there  corresponds  with  the  kinetic  energy 
of  the  molecules.  Suppose,  for  simplicity,  the 
molecules  or  atoms  are  all  moving  at  the  same 
speed,  and  their  total  kinetic  energy  is  the  same 
as  in  the  perfect  gas,  with  its  molecules  at  various 
speeds,  the  equivalent  uniform  speed  may  he  called 
the  "  equivalent  speed."  It  is  not  the  mean  speed, 
because  kinetic  energy  varies  as  the  square  of  the 
speed.  Then  if  the  gas  is  heated  at  constant 
volume,  the  entropy  varies  as  the  logarithm  of  the 
equivalent  kinetic  energy,  or  of  the  square  of  the 
equivalent  speed.  In  this  case  the  entropy  increases 
as  a  function  of  the  heat  of  the  gas.  Suppose  now 
the  gas  is  expanded  isothermally,  its  internal  energy 
and  equivalent  speed  remain  the  same  as  before, 
but  the  entropy  increases  as  the  logarithm  of  the 
volume.  This  has  nothing  to  do  with  the  kinetic 
energy  of  the  molecules  or  equivalent  speed,  as  they 
are  not  altered.  What  is  altered  is  the  infrequency 
of  collision  and  the  mean  path,  both  in  the  same 
sense  and  to  the  same  degree.  The  entropy  is  thus 
proportional  to  the  logarithm  of  the  length  of  the 
equivalent  free  path  of  the  molecules.  Thus  when 
a  perfect  gas  is  compressed  adiabatically,  its  entropy 
is  not  increased,  though  it  gets  hot,  as  the  increase 
of  the  logarithm  of  the  square  of  the  equivalent 


PHYSICAL   MEANING    OF   ENTROPY.     107 

speed  is  exactly  balanced  by  the  decrease  of  the 
logarithm  of  the  equivalent  free  path. 

In  heating  a  substance  which  undergoes  no 
physical  change,  such  as  a  solid,  the  entropy-  is 
again  proportional  to  the  logarithm  of  the  square 
of  some  sort  of  equivalent  molecular  speed.  When 
the  solid  melts,  or  a  liquid  vaporises,  there  appears 
to  be  some  change  of  kinetic  energy  by  regrouping 
the  parts  of  the  moving  systems  so  that  they  still 
communicate  the  same  energy  to  neighbouring 
bodies  by  battering  them,  but  have  some  addi- 
tional energy,  such  as  of  rotation,  which  is  not  com- 
municated by  battering.  The  entropy  increase  of 
physical  change  is  thus  still  proportional  to  the 
logarithm  of  the  square  of  an  equivalent  speed. 

One  reason  why  there  is  obscurity  as  to  the 
physical  meaning  of  entropy  is  that  it  is  treated  in 
orthodox  thermodynamics  as  one  quantity.  Some 
heterodox  observations  here  may  do  no  harm. 
Entropy  is  really  the  sum  of  several  quantities. 
In  the  case  of  a  pound  of  a  perfect  gas  increasing 
its  entropy  by  being  heated  at  constant  volume,  the 
increase  of  entropy  is  proportional  to  the  logarithm 
of  the  equivalent  momentum  of  the  particles.  That 
is  one  distinct  quantity,  which  I  have  called  ^,  or 
the  quantity  factor  of  sensible  heat.  If  a  pound  of 
ice  is  melted,  the  increase  of  entropy  is  again  the 
logarithm  of  some  equivalent  momentum.  This 
quantity  I  have  called  xp,  or  the  quantity  factor  of 


108  CONDUCTION. 

latent  physical  heat.  What  is  often  called  the  dis- 
gregation  energy  of  evaporating  water,  that  is  to 
sa}r,  the  increase  of  internal  energy  not  including 
any  external  work  done,  is  thus  OXP*  If  chemical 
work  is  done  by  taking  in  heat,  its  quantity  factor  is 
yc,  where  \^  ig>  again,  the  logarithm  of  some  equi- 
valent momentum  of  the  particles  forming  the  new 
combination.  Return  now  to  the  perfect  gas,  and 
let  it  expand  isothermally.  It  increases  in  volume, 
and  the  increase  of  entropy  is  proportional  to  the 
logarithm  of  the  volume.  We  thus  have  XA>,  X]>>  Xr> 
and  a  function  of  the  expansion  which  may  be  called 
e,  which  is  proportional  to  the  logarithm  of  the 
volume.  The  increase  of  entropy  is  the  sum  of 
these  four,  or 

<j>  =  x-s>  +  X/>  +  X*  +  €- 

We  need  not  consider  chemical  changes  here,  so 
we  have 

fe  d  4  =fe  d  x*  +fo  d  Xp  +fo  dc. 

Uncompensated  entropy  thus  occurs  in  engineer- 
ing in  two  ways.  If  heat  is  conducted  from  a  hot 
to  a  cold  body  the  bodies  average  their  energy  up. 
This  means  increasing  the  equivalent  momentum  of 
their  particles.  Thus,  if  a  shot  going  at  2,000  feet 
per  second  and  another  at  1,000  were  to  collide  and 
glance  off  at  equal  velocities,  so  that  the  total 
energy  remains  constant,  as  the  initial  energies 
are  as  4  to  1,  the  final  energies  are  2J  times  that 


PHYSICAL   MEANING    OF   ENTROPY.     109 

due  to  1,000  feet  per  second,  or  the  speed  is  as  the 
root  of  2J,  or  nearly  1,600  feet  per  second,  instead  of 
1,500,  which  would  be  the  average  speed.  The  mo- 
menta before  collision  were  as  2  and  1,  the  sum  being  3, 
whereas  after  collision  the  sum  is  in  proportion  of  3'2. 
In  averaging  up  their  energy  by  conduction  of  heat 
there  is  thus  an  increase  of  equivalent  momentum, 
and  the  logarithm  of  this  is  the  increase  of  x*  or  of 
<£,  as  far  as  conduction  goes.  This  increase  is 
obviously  uncompensated,  corresponding  to  an 
irreversible  change.  It  is  clearly  impossible  to 
reduce  this  equivalent  momentum  by  making  one 
part  hot  and  the  other  cold  again  without  taking  in 
heat  at  a.  high  temperature  and  giving  it  out  at  a 
lower,  that  is  to  say,  the  change  incurs  waste. 

When  the  perfect  gas  expands  isothermally,  whether 
it  does  external  work  or  not,  it  is  only  changed  as 
regards  the  free  path  of  its  particles,  but  to  bring  it 
back  to  its  original  state  needs  compression  by 
external  work,  and  if  a  particle  rebounds  from  a 
wall  that  is  approaching  it,  it  increases  its  velocity, 
so  that  if  the  compression  is  to  bring  the  gas  back 
to  its  original  state,  heat  must  be  given  out  and 
work  taken  in,  that  is  to  sa}r,  there  must  be  degra- 
dation of  work  into  heat,  of  which  a  portion  must 
be  waste. 

Thus  x  and  e  have  this  in  common,  that  their 
increase  means  that  to  get  the  substance  back  to  its 
standard  state,  even  reversibly,  involves  giving  out 


110  CONDUCTION. 

heat.  When  a  gas  expands  isentropically  its  increase 
of  c  is  exactly  balanced  by  its  decrease  of  x>  as  it 
cools,  and  the  entropy  remains  constant.  Splitting 
entropy  up  into  the  sum  of  a  number  of  quantities 
which  have  nothing  in  common  except  their  relation 
to  waste  is  new  and  unorthodox ;  but  if  the  reader 
wants  to  have  a  physical  idea  of  entropy  I  believe  it 
is  quite  necessary.  In  thermodynamics  the  entropy 
is  a  sum  of  quantities  having  only  one  property 
in  common,  and  has  therefore  no  single  physical 
meaning  except  as  regards  the  results  of  its 
increase. 

It  may  be  urged  that  it  is  highly  artificial 
for  x  and  c  and  their  sum  <£  to  vary  as  the 
logarithm  of  the  equivalent  momentum,  the 
logarithm  of  the  volume  or  free  path,  and  so  on ; 
whereas  volume  and  momentum  are  factors  of 
energy.  The  anstfer  is  that  we  have  by  a  pure 
accident  taken  temperature  so  that  it  varies  as  the 
square  of  the  equivalent  velocity  of  the  particles. 
If  we  take  as  the  tension  factor  T,  so  that  T  =  Jt9 
and  TT  as  the  corresponding  factor  of  heat,  we 
get  KB,  irj)t  and  ire,  varying,  not  with  the  logarithms 
of  momentum,  but  with  the  equivalent  momentum  ; 
and  i],  corresponding  with  «  for  the  expansion  func- 
tion, which  varies  directly  as  the  volume,  and  not  as 
its  logarithm.  Splitting  up  heat  into  factors,  and 
abolishing  the  idea  of  entropy  except  as  the  sum  of 
the  quantity  factors  and  the  expansion  function 


FACTORS    OF   HEAT.  Ill 

will,  I  feel  convinced,  make  the  study  of  thermo- 
dynamics more  simple. 

Conclusion. — Though  everything  in  this  book  is, 
I  believe,  consistent  with  modern  thermodynamics, 
the  whole  method  of  treatment  is  practically 
new.  It  has,  as  far  as  possible,  been  pointed  out 
when  anything  may  be  unorthodox,  so  that  the 
reader  who  is  new  to  the  subject  may  be  on  his 
guard. 

As  far  as  possible,  I  have  tried  to  put  a  difficult 
matter  in  plain  English,  and  not  in  mathematical 
symbols.  This  is  a  very  thankless  attempt,  for  if 
one  were  to  succeed  in  making  a  difficult  subject 
appear  clear  and  simple,  the  reader  would  think  the 
subject  itself  easy,  or  that  it  has  only  been  treated 
in  a  very  elementary  way,  and  very  often  that  the 
writer's  knowledge  is  also  equally  elementary, 
perhaps  untrustworthy.  The  sa"me  subject  treated 
in  an  involved  way,  with  a  quantity  of  obscure 
mathematics — a  first-rate  man's  mathematics  are 
always  clear,  however,  and  his  writing  as  simple  as 
the  case  allows — is  supposed  to  be  more  advanced 
and  deeper,  and  to  have  much  more  in  it. 

Few  realise  that  ease  of  reading  varies  inversely 
about  as  the  square  of  easy  writing,  and  if  one 
wants  kudos,  the  right  thing  is  to  make  the  subject 
as  complicated  and  difficult  as  possible,  and  espe- 
cially to  use  as  many  mathematical  symbols  as  can  be 
crowded  in.  It  is  quite  the  correct  thing  in  science, 


112  CONDUCTION. 

after  making  calculations,  to  "  remove  the  scaffold- 
ing," that  is  to  say,  to  publish  without  giving  the 
intermediate  steps,  so  that  others  should  think  the 
feat  greater. 

In  dealing  even  with  an  old  subject  in  a  novel 
way,  which  involves  criticism  of  existing  methods, 
it  is  necessary  for  a  writer  to  rely  on  his  own 
reasoning  from  step  to  step,  as  he  cannot  rest  on 
others  whose  way  is  different,  and,  in  his  opinion, 
inaccurate.  It  is  therefore  most  likely  there  are 
many  slips  and  false  steps  in  this  essay.  I  am 
not  young  enough  to  be  at  all  infallible  ;  one  would 
have  to  be  ver}r  young  indeed  to  be  infallible  in 
thermodynamics,  which  is  perhaps  the  most  slippery 
branch  of  science  there  is.  It  is  hoped  anyone  who 
notices  any  slips  will  write  to  me,  and  advantage 
will  be  gratefully  taken  of  any  corrections  or  sug- 
gestions, if  a  second  edition  should  be  demanded. 


APPENDIX. 


THE    REVERSIBILITY   OF  THERMO- 
DYNAMICS. 

A  NEW  way  of  looking  at  an  old  subject  can  do  no 
harm,  and  may  do,  and  generally  does,  a  great  deal 
of  good.  In  this  appendix  it  is  submitted  that  the 
ordinary  discussion  of  the  science  of  thermody- 
namics makes  the  subject  unnecessarily  obscure,  and 
leads  to  all  sorts  of  errors  ;  and  my  object  is  to  urge 
a  different  treatment  which,  it  is  hoped,  will  make 
the  science  more  easily  understood. 

There  is  no  branch  of  physical  science  so  com- 
monly misunderstood,  not  only  by  students,  but 
also  by  scientific  men  of  considerable  standing,  as 
thermodynamics.  Some  time  ago  I  asked  an 
eminent  authority  on  the  Continent  to  join  in  a  dis- 
cussion on  entropy  which  had  arisen  out  of  a  note 
to  the  presidential  address  I  had  the  honour  of 
delivering  before  the  Institution  of  Electrical  En- 
gineers. He  replied  that  though  I  was  quite  right 
in  my  statement  that  most  writers  on  physics  had 
got  hold  of  a  wrong  idea  of  entropy,  the  matter 
was  merely  pedagogic,  and  therefore  he  would  not 

E.  I 


114  APPENDIX. 

discuss  it  in  print.  A  question  of  pedagogy  would 
be  outside  the  province  of  an  engineer  ;  but  as  it  is 
not  really  a  question  of  method  of  teaching  young 
men  at  colleges,  but  of  giving  clear  ideas  to  scientific 
men  who  are  not  specialists  in  thermodynamics,  the 
subject  is  of  real  scientific  importance  ;  and  I  will- 
ingly acceded  to  the  request  of  the  Committee  of  the 
Mathematical  and  Physical  Section  of  the  British 
Association  to  open  a  discussion  on  thermodynamics, 
and  this  appendix  consists  essentially  of  the  paper 
written  for  that  purpose. 

Present  or  Orthodox  Treatment. — The  study  of 
thermodynamics  suffers  from  the  historical  develop- 
ment of  the  science.  During  the  first  half  of  last 
century  the  principle  of  the  conservation  of  energy 
was  on  its  trial,  and,  though  he  soon  discovered  his 
mistake,  Carnot  had  supposed  that  in  his  cycle  as 
much  heat  came  out  of  the  working  substance  as 
went  in.  I  pretend  to  no  knowledge  of  the  history 
of  thermodynamics,  but  it  looks  as  if  Clausius, 
realising  that  the  heat  is  not  conservative  in  a  Carnot 
cycle,  found  a  function  S,  which  is.  Then  came  the 
question  of  the  second  law  arid  the  best  way  of 
formulating  it,  and  the  whole  discussion  centred 
round  the  Carnot  cycle  with  dQ/T  a  complete  dif- 
ferential. Instead  of  S,  Q  and  T,  I  have  used  3>,  H 
and  6  in  this  book,  as  Q  is  generally  used  in  other 
branches  of  physics  for  the  quantity  factor  of  energy, 
not  for  energy,  and  T  for  kinetic  energy ;  and  if  $  is 


ORTHODOX   TREATMENT.  115 

used  for  entropy,  0  may  well  be  used  for  tempera- 
ture, as  by  Rankine,  Maxwell  and  others. 

The  first  matter  generally  is  to  state  the  law  of 
the  conservation  of  energy  and  to  define  "  heat." 
Here  there  is  a  complete  failure.  The  definitions 
and  explanations  really  come  down  to  saying  in  a 
more  or  less  roundabout  and  complicated  way  that 
heat  is  what  makes  things  hot.  In  its  way  this 
would  be  quite  a  nice  and  all-sufficing  definition, 
but  it  is  not  adhered  to  at  all.  The  domination  of 
a  name  comes  in.  The  increase  of  internal  energy 
necessary  to  produce  a  change  of  (physical)  state, 
such  as  melting  ice,  was  called  heat  before  thermo- 
dynamics began.  Objectors  to  the  caloric  theory,  I 
think,  refused  to  call  this  energy  heat,  in  order  to 
annoy  their  opponents  ;  but  when  the  caloric  advo- 
cates disappeared  by  dying  out,  or  perhaps  even  by 
being  occasionally  convinced,  increase  of  internal 
energy  became  latent  heat  again ;  and  then  the 
increase  of  internal  energy  and  external  work 
together  became  "latent  heat,"  having  a  symbol 
Cp  or  Kp  to  itself,  so  that  increase  of  U  where  there 
is  no  chemical  change  is  heat,  and  the  external 
work  done  per  degree  rise  of  temperature  is  part  of 
the  latent  heat  of  the  body,  so  the  definition  goes  by 
the  board,  and  heat  that  makes  things  hot  is  called 
"  sensible,"  perhaps  as  a  stigma  on  the  other  kinds 
of  heat.  The  definition  of  heat  is  thus  wanting.  I 
will  quote  from  some  correspondence  I  had  with  one 

i2 


116  APPENDIX. 

of  our  leading  mathematical  authorities  on  thermo- 
dynamics. "  What  is  wanted  is  a  definition  of  heat. 
Can  you  define  heat  ?  If  you  can  give  a  definition 
which  will  make  it  perfectly  clear  what  is  heat  and 
what  is  not  heat,  you  will  be  doing  good  work.  But 
you  must  cover  all  possible  cases."  The  italics  are 
his.  Now  what  does  my  correspondent  want  ? 
There  are  two  classes  of  definition  in  science.  One 
is  a  mere  verbal  distinction  which  fits  pre-existing 
ideas  and  conveys  no  information.  For  instance, 
everyone  of  my  readers  has,  no  doubt,  a  perfectly 
clear  idea  of  what  energy  is,  and  also  knows  what  is 
energy  and  what  is  not,  in  any  given  case ;  but  to  define 
energy  so  as  to  convey  any  information  to  anybody 
who  has  no  pre-existing  notion  of  it  is  no  easy  task. 
But  perhaps  what  he  really  meant  was  something  like 
this.  "  Is  latent  heat,  heat  ?  Surely  external  work 
is  not  heat.  Is  chemical  energy  heat  ?  How  far  is 
radiation,  thermal  or  luminous,  heat  ?  Are  electron 
flights  ever  heat  ?  If  so,  when  are  they  kinetic  energy, 
and  when  heat  ?  When  are  electro-magnetic  waves 
heat  ?  Is  the  energy  of  pedesis  heat  ?  If  so,  and 
the  little  particles  are  made  to  jolt  bigger  particles, 
and  so  on  till  they  finally  shake  lumps,  is  that  heat 
or  directed  kinetic  energy,  and  where  is  the  line  of 
demarcation  ?  At  what  stages  in  the  electrostatic 
or  electro-magnetic  hysteresis  cycles  does  the 
entropy  increase  ? "  I  do  not  know  whether  a 
definition  which  would  answer  these  questions  would 


ORTHODOX   TREATMENT.          117 

satisfy  my  friend,  but  some  such  definition  is 
wanted.  Many  of  these  questions  are,  no  doubt, 
answered  by  inference  in  papers  on  those  particular 
subjects  written  from  a  thermod}7namical  standpoint; 
but  an  ordinary  treatise  on  thermodynamics  does 
not  touch  them,  or  answer  my  friend's  question  in 
the  least. 

Then  there  is  a  difficulty  about  thermometry. 
The  Fahrenheit  scale  takes  the  expansion  of  mercury 
one-hundredth  per  cent,  as  a  "  degree,"  the  Centi- 
grade divides  the  difference  of  two  arbitrary  tempera- 
tures into  100  parts,  and  is  therefore  regarded  as 
more  scientific,  and  the  Reaumur  exists  in  real  life 
but  not  in  science.  Much  trouble  is  therefore  devoted 
to  elucidating  Kelvin's  scale,  and  it  is  then  gene- 
rally assumed  that  the  Fahrenheit  and  Centigrade 
thermometers  read  in  the  Kelvin  scale,  only  with 
different  zeros  and  different  units  of  temperature. 
The  main  point  to  which  attention  is  especially 
directed,  however,  is,  that  writers  are  so  eager 
about  showing  that  production  of  work  means  dis- 
appearance of  heat,  and  so  anxious  to  explain 
Kelvin's  scale  and  to  discuss  the  second  law  and 
Carnot's  cycle,  that  the}r  discuss  the  whole  subject 
only  in  connection  with  ideal  reversible  changes. 
Then  the  fact  of  dH/0  being  a  complete  differential 
in  reversible  changes  is  very  attractive  to  many 
people,  because  they  can  then  get  a  lot  of  un- 
deserved happiness  in  stringing  together  partial 


118  APPENDIX. 

differential  equations  tracing  the  relations  of  every 
conceivable  thermoclynamic  quantity  with  every  other, 
down  to  the  most  remote  cousinships.  Thermo- 
dynamics is  thus  apt  to  degenerate  into  nothing 
better  than  an  exercise  in  differential  equations. 

In  discussing  the  principle  of  entropy,  the  entropy 
of  the  working  substance  is  treated  almost  exclu- 
sively, and  that  in  connection  with  reversible  changes 
and  cycles  alone.  Irreversible  changes  are  dis- 
cussed as  a  sort  of  curious  exception,  but  only  very 
slightly.  In  many  cases  two  bodies  are  considered 
before  and  after  equalising  their  temperatures,  and 
their  increase  of  total  entropy  is  discussed,  but 
there  is  no  explanation  of  how  the  entropy  increases; 
only  the  initial  and  final  stages  are  considered. 

But  the  chief  ground  of  the  accusation  I  have  the 
honour  to  bring  is  that  anxiety  to  discuss  the  Carnot 
cycle  and  its  bearing  on  the  second  law,  and  to  work 
with  a  quantity  4>  which  could  be  assumed  conserva- 
tive, has  led  to  a  very  elaborate  mathematical 
development  of  the  theory  of  reversible  changes, 
and  has  relegated  real  thermodynamics  to  an 
undeserved  background. 

My  historical  notions  may  be  wrong,  but  it  looks 
as  if  Clausius,  when  Carnot's  reasoning  had  been 
vitiated  by  his  original  mistake  as  to  conservation  of 
heat,  looked  for  something  that  was  conservative  in 
the  Carnot  cycle  and  found  entropy.  He  got  the 
"  tropie  "  from  the  Greek,  and  put  on  the  "en" 


ORTHODOX   TREATMENT.          119 

to  make  the  word  sound  something  like  its  conserva- 
tive analogue  "  energie."  Then  in  discussing  real 
changes  he  found  that  entropy  is  not  conservative, 
and  founded  the  principle  of  modern  thermo- 
dynamics, that  the  entropy  of  the  universe  strives  to 
increase  ;  a  principle  which  is  the  very  backbone  of 
the  science.  In  spite  of  this,  the  ordinary  treatise, 
though  it  mentions  Clausius's  principle,  and  perhaps 
touches  on  Kelvin's  work  of  the  same  period,  says 
very  little  about  it.  By  "ordinary  treatise,"  I 
need  hardly  say,  I  do  not  mean  to  include  such 
works  as  those  of  Planck  or  Duhem,  or  those  written 
by  the  abler  modern  physical  chemists. 

Entropy  is  denned  with  reference  to  reversible 
changes  only  by  the  equations 

fdR/e  =  <*>  and  (/)dR/0  =  0, 

which  give  no  sort  of  notion  what  entropy  is,  and. 
which  are  only  numerically  correct  in  the  case  of  rever- 
sible changes  and  cycles  respectively,  and,  in  fact, 
are  never  even  numerically  true.  The  foundation 
of  thermodynamics  is  that  neither  of  these  equations 
is  true.  I  know  of  no  writer  who  has  tried  to  give 
any  sort  of  explanation  of  what  is  meant  by  entropy, 
except  that  it  is  the  quantity  factor  of  heat,  which  is 
obviously  nonsense.  It  is  not  meant  that  specialists 
on  thermodynamics  do  not  understand  their  own 
subject  or  write  inaccurately ;  my  complaint  is  that 
the  treatment  of  thermodynamics  is  obscure  and 


120  APPENDIX. 

misleading,  and   has   lead  to  great  confusion  and 
much  error. 

Nearly  all  writers  on  physics,  for  instance,  define 
entropy  by  the  equation  dH/6  =  d®,  or  its  equivalent. 
That  is  to  say,  they  are  led  to  suppose  that  what 
would  be  true  if  there  were  any  reversible  changes 
is  true  in  fact.  dH/0,  or  d®,  is  treated  as  a  typical 
complete  differential.  An  adiabatic  change  is  sup- 
posed to  be  isentropic.  (y*)cZH/0  =  0  is  supposed 
to  be  the  second  law  of  thermodynamics.  Entropy 
is  said  to  be  a  factor  of  heat,  and  so  on,  a  whole 
tissue  of  errors  and  misconceptions  thus  arising 
from  the  impression  given  by  looseness  of  diction. 
When  any  man  of  ordinary  intelligence  and  the 
necessary  training  cannot  understand  what  has  been 
written,  it  may  be  taken  for  granted  it  is  the  fault 
of  the  writer.  Generally,  people  are  of  an  opposite 
opinion,  perhaps  thoughtlessly,  and  if  the  treatment 
of  a  subject  proves  to  be  puzzling,  the  author  is 
rather  to  be  admired  for  being  master  of  such  a 
difficult  subject,  and  not  blamed  for  not  making  it 
simple ;  whereas,  if  he  put  the  matter  simply  and 
clearly,  he  would  get  but  little  credit  for  his  work. 
This  is  especially  true  of  subjects  that  can  be 
treated  mathematically ;  the  scientific  world  has  an 
immense  reverence  for  anything  put  in  mathematical 
language,  though  it  needs  much  more  ability  and  a 
clearer  head  to  put  it  in  words.  Was  it  not  Maxwell 
who  said  that  mathematics  is  a  shorthand,  and  that 


ORTHODOX   TEEATMENT.  121 

anything  that  can  be  put  in  symbols  can  be  put  in 
words,  if  the  writer  really  understands  it  ?  Nobody 
really  understands  unless  he  can  express  himself  in 
words.  Thermodynamics  seems  to  be  peculiarly 
unfortunate  in  being  a  vehicle  for  blind  mathematics. 
It  is  not  for  a  moment  hinted  that  science  cannot 
be  clear  if  it  is  mathematical ;  quite  the  reverse.  The 
great  writers  on  physics  use  mathematics  in  such  a 
way  that  they  are  merely  shorthand  methods  of 
explaining  physical  ideas.  Any  reader  grasps  the 
physical  ideas  at  once,  unless  his^mathematical  equip- 
ment is  too  limited  ;  but  even  then  he  feels  the 
physical  ideas  are  there,  and  very  often  finds  the  study 
a  good  way  of  learning  mathematics.  On  the  other 
hand,  another  writer  will  make  the  simplest  matter 
unintelligible  by  mathematical  treatment.  Thus 
such  an  expresion  'dsfdH/0  =  $  is  a  perfectly  clear 
statement  of  fiction.  But  when  we  come  to  fact  we 
are  told  that^dH/0<$.  Now  what  does  that  mean  ? 
Being  an  engineer,  in  coming  across  such  an 
expression  I  sought  an  interpretation  that  gave  a 
physical  meaning.  0  could  not  be  the  temperature 
of  the  reservoir,  because  there  may  be  no  reservoir  ; 
for  instance,  the  body  ceding  heat  may  not  be  at 
uniform  temperature,  for  example,  in  considering 
part  of  a  conductor  of  heat  where  there  is  a  tempera- 
ture gradient.  Then  0  cannot  generally  be  the 
temperature  of  the  body  because  that  is  not  usually 
uniform.  To  make  sense,  0  is  the  temperature  of 


122  APPENDIX. 

the  bounding  surface,  and  H  is  a  flux  through  the 
surface.  But  I  find  I  appear  to  be  rather  peculiar 
in  this  interpretation.  What  then  does  the  orthodox 
writer  mean  by  H  and  0  in  irreversible  changes,  and 
what  idea  is  really  in  the  mind  of  the  reader  as  he 
glides  over  such  expressions?  That  there  is  a 
tendency  towards  blind  mathematics  and  fogginess 
of  idea  in  thermodynamics  is,  I  submit,  proved  by 
the  inaccuracy  of  the  terms  used,  the  absence  of 
named  units,  and  the  absence  of  physical  ideas. 
How  many  chemists  have  any  clear-cut  idea  of 
what  the  "  Thermodynamic  Potentials  "  really 
are? 

The  entropy  principle  was  formulated  half  a 
century  ago.  Before  and  since  then  the  question 
of  what  determines  the  direction  of  chemical  action 
was  eagerly  discussed.  The  entropy  principle  was 
supposed  to  be  familiar  to  every  scientific  man,  but 
it  was  not  assimilated  in  the  least.  After  about 
twenty  years  the  chemical  puzzle  was  solved  by  Horst- 
mann,  who  pointed  out  that  the  increase  of  entropy 
is  the  criterion.  Bayleigh,  doubtless  independently, 
said  the  same  in  1875  ;  then  followed  Massieu, 
Gibbs,  and  Helniholtz.  Rayleigh  said  it  in  plain 
English ;  the  others,  especially  Massieu,  said  it  in 
mathematics.  But  even  that  has  produced  but 
little  effect,  and  now,  after  more  than  half  a  century, 
the  principle  of  increase  of  entropy  is  by  no  means 
generally  understood  by  chemists,  nor  by  engineers, 


OKTHODOX   TREATMENT.  123 

though  they  are  both  exceedingly  anxious  to  under- 
stand and  use  thermodynamics.  This  might  be 
the  fault  of  chemists,  engineers,  and  other  scientific 
men,  but  it  seems  much  more  likely  to  be  due  to 
obscurities  in  thermodynamics. 

It  is  easy  to  get  the  reputation  of  a  reformer  by 
finding  fault  with  existing  conditions.  It  is  more 
to  the  purpose  to  show  what  one  thinks  is  an 
improvement  by  a  specimen.  I  have,  therefore, 
attempted  a  short  elementary  exposition,  especially 
as  far  as  it  concerns  mechanical,  apart  from  chemical, 
engineers.  This  forms  the  body  of  this  book.  In 
this  is  exemplified  a  treatment  of  elementary  thermo- 
dynamics, which  depends  largely  on  the  reversal  of 
the  ordinary  method.  For  instance,  dissipation  and 
waste  are  explained  before  reversibility  and  the 
Carnot  cycle  ;  entropy  is  defined  without  any  refer- 
ence at  first  to  heat  passed  into  a  body ;  the  increase 
of  entropy  is  treated  as  normal,  and  reversible 
changes  as  a  purely  ideal  exception  ;  the  second  law 
of  thermodynamics  is  merged  in  the  impossibility  of 
perpetual  motion ;  the  entropy  of  the  universe  and 
of  the  isolated  system  is  treated  before  that  of  the 
working  substance ;  and  the  growth  of  entropy 
during  conduction  is  treated  later.  For  that  reason 
this  essay  is  headed  as  it  is.  To  put  the  matter 
succinctly,  for  easy  grasp,  a  sort  of  sketch  will  now 
be  set  out,  which  also  summarises  the  body  of  the 
previous  chapters. 


124  APPENDIX. 


SKELETON    OF    THERMODYNAMICS. 

Energy  can  be  divided  broadly  and  clearly  into 
two  kinds,  which  may  be  called  work  and  heat. 
Different  kinds  of  work  can  be  converted  one  into 
another  wholty  in  idea,  and  nearly  wholly  in  fact ; 
different  kinds  of  heat  can  be  converted  wholly  into 
one  another.  Work  can  be  wholly  converted  into 
heat,  but  heat  can  only  be  partially  converted  into 
work  if  the  converting  substance  returns  to  its 
original  state,  the  rest  of  the  heat  being  degraded 
into  a  less  available  form.  This  definition  of  heat 
includes  the  heat  that  makes  things  hot,  and  loco- 
motive heat  in  general,  and  it  also  includes  "latent 
heat  "  at  constant  volume,  but  only  part  of  any  mis- 
named "latent  heat"  that  includes  any  form  of 
external  work.  It  includes  latent  heat  of  fusion,  of 
vaporisation  apart  from  external  work,  and  of  allo- 
tropic  modification.  AVliat  is  most  heterodox  is 
that  it  includes  chemical  energy. 

There  are  three  classes  of  perpetual  motion. 
The}r  are  all  impossible,  and  two  of  them  are  not 
approximately  attainable,  but  third-class  perpetual 
motion  is  approximately  possible.  / 

First-class  perpetual  motion  is  when  an  otherwise 
isolated  system  can  give  out  energy  continually, 
or  without  decreasing  its  own.  The  impossibility 
of  first-class  perpetual  motion,  coupled  with  the 
impossibility  of  an  energy  sink  is  the  principle 


DEFINITION    OF   ENTKOPY.         125 

of  conservation  of  energy,  and  gives  the  first  law  of 
thermodynamics . 

Second-class  perpetual  motion  is  when  an  isolated 
system,  in  spite  of  friction  or  its  equivalent,  goes  on 
continually.  Second-class  perpetual  motion  would 
not  contradict  the  principle  of  the  conservation  of 
energy.  Its  impossibility  is  the  second  law  of 
thermodynamics. 

Third-class  perpetual  motion  is  when  an  isolated 
system — a  mechanism,  for  instance — has  no  friction, 
or  no  equivalent  of  it,  and  therefore  goes  on  or 
changes  continually.  The  impossibility  of  third- 
class  perpetual  motion  is  the  third  law  of  thermo- 
dynamics. It  is  not  called  the  third  law,  but  it 
deserves  that  rank. 

Waste. — Work  is  valuable  to  man,  and  the  portion 
of  any  heat  that  can  be  converted  into  work,  leaving 
the  converting  substance  in  its  original  state,  is 
valuable  ;  but  the  rest  of  the  heat  is  waste.  When 
work  is  degraded  into  heat,  a  portion  of  it  is  avail- 
able as  work,  the  rest  is  waste.  The  third  law  of 
thermodynamics  states,  in  other  words,  that  no 
change  in  nature  takes  place  without  incurring  waste. 

Definition  of  Entropy. — The  increase  of  entropy 
of  an  isolated  system,  multiplied  by  the  lowest 
available  temperature,  is  the  incurred  waste.  The 
waste,  therefore,  depends  on  the  lowest  available 
temperature  and  the  entropy.  As  any  given  change 
can  only  increase  the  lowest  available  temperature 


126  APPENDIX. 

infinitesiinally,  the  second  and  third  laws  may  be 
stated  in  the  form,  "  The  entropy  of  the  universe 
strives  to  increase."  So  does  the  waste.  Its  maxi- 
mum will  be  reached  when  all  heat  is  unavailable  and 
there  is  no  work  left.  The  incurred  waste  increases 
faster  than  the  entropy  of  the  universe,  because 
the  lowest  available  temperature  is  always  rising. 
The  limiting  expression  "  incurred  "  is  used  before 
waste,  advisedly.  Suppose,  in  an  isolated  system, 
some  gas  expands  doing  some  work  external  to  the 
gas.  The  heat  of  the  isolated  system  is  diminished, 
the  work  increased,  and  the  total  energy  unchanged. 
The  heat  being  diminished  without  any  correspond- 
ing rejection  at  lower  temperature,  the  waste  portion 
is  decreased.  There  is  thus  actual  decrease  of 
waste.  But,  to  get  the  gas  back  to  its  original 
state,  work  must  be  degraded  into  heat  again,  and, 
according  to  the  third  law,  there  is  then  increase  of 
waste.  Had  the  process  been  reversible,  there 
would  have  been  no  increase  of  incurred  waste  in 
the  system,  but,  as  far  as  the  gas  alone  is  concerned, 
its  own  increase  of  entropy  multiplied  by  the  lowest 
available  temperature  would  be  the  waste  involved 
in  bringing  it  back  to  its  original  state. 

When  during  a  change  the  entropy  of  a  body 
forming  part  of  the  system  diminishes,  the  entropy 
of  another  part  must  increase  to  the  same  extent 
in  an  ideal  change,  and  to  a  greater  extent  in  a 
real  change.  In  the  ideal  change  the  increase  of 


DEFINITION   OF   ENTROPY.         127 

entropy  of  the  body  is  called  compensated,  in  a  real 
change  the  part  of  the  increase  equal  to  the 
corresponding  decrease  elsewhere  is  compensated 
entropy,  <&•,  and  the  balance  is  uncompensated 
entropy,  ®u,  where  &u  =  Q  —  fdH/Q. 

When  the  heat  of  one  body  diminishes  as  that  of 
another  in  contact  with  it  increases,  it  is  usual  to 
say  that  the  heat  moves  from  one  to  the  other. 
Similarly,  when  the  entropy  of  one  body  diminishes 
while  that  of  another  in  contact  increases  to  the 
same  or  a  greater  extent,  it  is  convenient  to  say 
the  entropy  moves  and  grows  also.  From  the  idea 
of  the  entropy  of  an  isolated  system  and  its  increase 
in  every  change,  it  is  easy  to  pass  to  the  idea  of  the 
increase  or  decrease  of  the  entropy  of  a  body.  In 
the  case  of  reversibility  the  entropy  of  the  isolated 
system  is  conservative,  so  that  the  discussion  of 
the  behaviour  of  any  body  or  bodies  may  be  treated 
as  in  dynamics,  by  examining  the  changes  and  inter- 
relations of  the  external  co-ordinates ;  and  the 
conservation  of  entropy,  in  the  ideal  case  of  reversi- 
bility, is  a  very  convenient  assumption,  especially 
as  it  specially  helps  the  treatment  as  in  dynamics, 
the  conservation  of  entropy  corresponding  with  the 
conservation  of  energy.  The  exaggeration  of  the 
importance  of  this  treatment  has  done  much  to 
foster  incomplete  understanding  of  thermodynamics. 

By  treating  the  increase  of  entropy  of  an  isolated 
svstern  as  the  fundamental  idea  and  variations  of 


128  APPENDIX. 

the  entropies  of  the  parts  as  derived  ideas  it  is  sub- 
mitted that  a  much  clearer  notion  is  obtained.  It 
is  more  convenient  in  practice,  however,  to  consider 
the  entropy  of  some  substances  or  volumes  which 
form  parts  of  isolated  systems.  In  this  connection 
we  have  the  following  propositions  : — 

Reversibility. — A .  reversible  change  is  an  ideal 
change  which  could  take  place,  with  everything 
involved,  in  the  other  direction. 

When  a  reversible  change  takes  place  in  an 
isolated  system  there  is  no  increase  of  incurred 
waste,  for  if  there  were,  on  reversing  there  would 
be  decrease  of  incurred  waste,  and  second-class 
perpetual  motion  would  be  possible. 

When  a  reversible  change  takes  place,  therefore, 
the  entropy  of  the  isolated  system  remains  constant. 
\A  reversible  cycle  is  a  series  of  reversible  changes 
which  brings  the  working  substance  back  to  its 
original  condition.  The  entropy  of  an  isolated 
system  is  not  altered  by  a  reversible  cycle  being 
performed  inside  it. 

In  any  reversible  change,  the  total  entropy  of  the 
system  being  constant,  the  increase  of  entropy  of 
any  part  is  compensated  by  an  equal  decrease  of 
another  part.  The  decrease  of  entropy  of  any  body 
can  only  take  place  by  a  process  which  involves  at 
least  equal  increase  of  entropy  of  the  rest  of  an 
isolated  system.  Thus  the  entropy  of  a  body  may 
be  reduced  without  its  giving  out  heat  b}r  a  therm o- 


REVERSIBILITY.  129 

electric  circuit,  but  that  involves  at  least  equal 
increase  of  entropy  elsewhere  in  the  electric  circuit. 
Except  in  the  case  of  a  thermo-electric  circuit,  the 
entropy  of  a  body  can  only  be  reduced  by  its  giving 
out  heat,  for  if  it  could  be  reduced  by  giving  out 
work,  there  would  be  no  corresponding  increase 
outside,  and  second-class  perpetual  motion  would 
be  possible. 

The  reversible  increase  of  entropy  of  a  body  can 
only  take  place  by  its  taking  in  heat  from  outside 
(except  by  a  thermo  circuit),  otherwise  the  entropy 
of  an  isolated  system  would  be  increased,  and  the 
change  would  not  be  reversible. 

The  entropy  of  a  body,  compared  with  its  entropy 
in  a  standard  state,  is  the  entropy  that  must  at  least 
come  out  of  it  in  bringing  it  back  to  its  standard 
state  ;  or  it  is  a  quantity  which,  when  multiplied  by 
the  lowest  available  temperature,  gives  the  waste 
portion  of  the  heat  given  out  in  bringing  the  body 
to  its  standard  state  by  reversible  changes. 

As  the  waste  portion,  or  loss  L,  of  heat  given  out 
through  the  envelope,  H,  is  H02/#i  where  6\  is  the 
temperature  of  the  envelope,  and  6%  the  lowest 
available  temperature;  and  as  by  definition  02$  =  L, 
<£  =  H/0i,  or  if  the  temperature  of  the  envelope  varies 
during  the  output  of  the  heat  H,  &=fdHI0.  Thus 
the  entropy  of  a  body  in  state  B  compared  with 
state  A  is  ®v  —  ®A=Jdfi/0,  where  H  is  the  heat  that 
would  be  given  out  or  taken  in  if  the  change  were 

E.  K 


130  APPENDIX. 

reversible.  The  third  law  shows  that  in  fact 
<l>»  —  3>A>ySH/#i,  the  excess  being  uncompensated 
entropy  &u. 

The  entropy  of  a  body  thus  depends  on  its  state, 
and  not  on  its  past  history. 

Equilibrium. — If  any  small  change  imagined  in 
an  isolated  system  involves  decrease  of  entropy, 
that  change  is  impossible,  in  accordance  with  the 
second  law.  If  it  causes  no  change  of  entropy,  there 
is  equilibrium  as  far  as  that  change  is  concerned,  and 
no  change  will  take  place  in  either  direction.  If  the 
change  would  increase  the  entropy,  the  change  is  pos- 
sible and  may  take  place,  but  only  in  that  direction. 

A  possible  change  thus  involves  increase  of 
entropy,  or  uncompensated  entropy. 

The  uncompensated  entropy  of  the  working  sub- 
stances can  be  expressed  in  terms  of  the  co-ordinates 
of  each  working  substance  alone,  without  other 
reference  to  the  rest  of  the  isolated  system.  Thus 
in  the  case  of  a  working  substance,  where  <£?,,  is  the 
uncompensated  entropy,  H  the  heat  taken  in,  6  the 
temperature  of  the  envelope,  U  the  heat  of  the  body, 
according  to  the  definition  given  above,  and  W  the 
external  work  done. 

d*u  =  d*  -  dH/0  and  dR  =  dU  +  dW, 

therefore  d*u  =  d*  -  dU/6  -  rZW/0, 
and  if  there  is  equilibrium, 


EQUILIBRIUM.  131 

If  there  is  uncompensated  entropy  on  making  the 
chane 


The  integral  of  this  quantity  is  therefore  written 
with  the  sign  reversed  and  disguised  under  the 
name  of  a  "  Thermodynamic  Potential,"  so  that 
increase  of  uncompensated  entropy  decreases  instead 
of  increasing  the  quantity.  Uncompensated  entropy 
is  a  drawback,  and  thermodynamic  potential  is  an 
advantage.  It  is  with  great  respect  to  Professor 
Planck  or  Professor  van  Laar  *  that  it  is  submitted 
that  thermodynamic  potential  is  a  bad  name  for  the 
function 


The  third  law  shows  that  heat  can  never  be 
changed  into  work  to  the  extent  it  would  be  by  a 
reversible  cycle.  The  work  that  would  be  gained 
by  a  reversible  cycle,  and  is  not  gained  in  a  real 
change,  may  be  called  the  "  ungained  work." 
Professor  Duhem  t  calls  it  the  "uncompensated 
work."  The  ungained  work,  W«=y&W>«,  where 
6  is,  if  the  working  substance  is  of  uniform 
temperature,  the  temperature  of  the  envelope  or 
body.  If  the  temperature  is  not  uniform,  6  is  the 
temperature  of  each  volume  element  in  which  the 
uncompensated  entropy  grows,  and  integration  is 

*  Or  whoever  is  responsible  for  the  expression. 
t  I  think  the  term  is  originally  due  to  Professor  Duhem  , 
but  my  memory  may  be  at  fault.     See  p.  70. 


132  APPENDIX. 

carried  out  throughout  the  volume  ;  but  as  the 
expression  is  important  only  when  Wu  =  0,  the 
meaning  of  6  otherwise  does  not  matter  for  the 
moment. 

If  a  small  change  involves  decrease  of  ungained 
work  it  is  impossible  ;  if  the  ungained  work  is  zero, 
the  substance  and  its  externals  are  in  equilibrium 
as  far  as  that  change  goes  ;  if  the  ungained  work  is 
positive  the  change  is  possible,  and  may  take  place 
in  that  direction  only.  As  the  ungained  work  does 
not,  unfortunate!}',  exist  outside  the  substance,  nor 
anywhere  else  as  work,  it  is  best  not  to  call  it  Ww, 
but  A  or  D.  Then,  if  the  ungained  work  is  zero 
during  a  proposed  change,  the  "  thermodynamic 
potential  "  is  said  to  remain  constant.  If  the  un- 
gained work  increases,  the  thermodynamic  potential 
is  said  to  decrease.  The  ungained  work  due  to 
a  small  change  may  be  represented  in  terms  of 
the  co-ordinates  of  the  working  substance. 


The  external  work,  W,  is  general.     It  may  be 
done  electrically,  or  by  expansion.     If  by  expansion 

<ZW  =pdv  =  dpi:  —  vdp. 
Then  d  A  =  6d$>  -  dU  +  dW  =  d6*  -  &dO  -  dU  +  dW 


(1) 

(2)  dt*  =-</U-<Z\V 


EQUILIBRIUM.  183 


(3) 

(4)  dp*  =0d&-dU-dp 

(5)  <fc,.A=,/0$-dU 

(6) 
(7) 
(8) 


If  the  various  partial  differentials  are  taken  as 
zero,  the  condition  of  equilibrium,  and  the  right- 
hand  terms,  integrated  without  adding  any  constants, 
and  reversed  in  sign  so  that  their  decrease  corre- 
sponds with  increased  or  ungained  work,  eight  more 
thermodynamic  potentials  are  involved.  Part  of 
No.  1  gives  what  is  called  "free  energy."  It  is  not 
energy,  and  is  in  no  way  remarkable  for  freedom  of  any 
kind.  The  results  of  three  of  these  are  the  Gibbs's 
functions,  which  have  been  called,  first  by  Professor 
Duhem,  I  believe,  thermodynamic  potentials,  no 
doubt  because  they  are  constant  for  equilibrium, 
like  potential  in  mechanics.  Again,  it  is  submitted 
with  much  deference,  that  the  name  is  misleading. 
If  the  signs  were  not  reversed,  stabilities,  or  stabilit}r 
functions,  might  be  a  better  name.  In  non-thermal 
reversible  mechanics,  equilibrium  or  maximum 
stability  is  reached  when  the  potential  is  a  minimum. 
Maximum  stability  can  then  be  called  minimum 
potential  ;  but,  surety,  it  does  not  follow  from 
this  that  maximum  stability  is  always  synonymous 
with  minimum  potential,  and  that  thermodynamic 


134  APPENDIX. 

potential  is  a  synonym  for  the  negative  of  stability, 
especially  when  applied  to  quantities  that  are  not 
energy,  though  of  the  same  dimensions  as  energy. 
Besides,  the  term  potential  is  getting  worn  out. 
Counting  the  eight  potentials  just  given  as  one 
specimen,  I  can  think  of  nine  kinds  of  potential  used 
in  physics,  and  a  collector  could  doubtless  find  many 
more.  They  have  various  meanings  and  dimensions. 
Four  or  five  of  them  belong  to  thermodynamics.  If 
we  have  very  many  more  potentials,  the  term  may 
become  a  little  difficult  to  define  accurately  and 
succinct!}7,  though  it  is  always  impressive. 

By  defining  entropy  first  in  terms  of  the  waste, 
and  dealing  with  the  increase  of  entropy  of  -an 
isolated  system  first,  and  coming  to  the  relation  of 
increase  of  entropy  to  heat  taken  in  reversibly  later, 
the  chance  of  confusing  entropy  withy^H/0  generally, 
and  supposing  entropy  is  a  factor  of  heat  is  avoided. 
This  prepares  the  way  for  discussing  the  real  factors 
of  heat,  such  that  ^/&ZX  =  U,  where  #  is  the  tension 
and  X  the  quantity  factor.  In  other  branches 
of  physics,  not  only  fadb  =  W,  where  a  and  b  are 
tension  and  quantity  factors,  but  where  c  is  capacity 
for  the  quantity,  so  that  db/da  =  c,  facda  —  W,  or 
when  c  is  constant,  W  =  Ja9c.  In  that  case  the 
tension  factor  is  \/0  =  T,  and  the  corresponding 
quantity  factor  H,  so  thatyrdII  =  U.  In  a  change 
where  there  is  no  external  work  done  and  no 
ungained  work  X  is  numerically  equal  to  3>,  and  is 


EQUILIBRIUM.  185 

the  logarithm  of  the  square  of  some  sort  of  equiva- 
lent velocity  of  the  particles.  Thus  in  the  case  of 
a  perfect  gas  the  entropy  depends  on  the  logarithm 
of  the  square  of  the  equivalent  speed,  and  the  loga- 
rithm of  the  equivalent  free  path  of  the  particles  of 
the  gas  if  there  is  one,  or  of  each  if  there  are  more. 

The  frequency  of  collision  has  to  do  with  external 
work,  and  X  depends  only  on  the  speeds.  The 
physical  meaning  of  r  and  n  is  much  simpler;  r 
varies  as  the  equivalent  speed,  and  n  as  the  equiva- 
lent momentum. 

Take  the  case  of  chemical  energy  as  one  of  the 
forms  of  heat.  U  can  be  divided  up  into  Us,  Up,  Uc 
for  sensible,  latent  physical  and  chemical  heat. 
f 6dTls  =  Us  ;  frdHc  =  Uc  and  so  on.  At  present 
heats  *  of  combustion  or  combination  are  given  in 
the  lump.  In  the  future  we  may  have  tables  giving 
the  6  and  x  or  the  r  and  TT  of  each  free  element,  and 
their  change  on  reaction  ;  or  even  their  0  and  </>. 
Then  we  would  know  beforehand  in  which  direction 
each  reaction  would  go.  The  question  of  the 
factors  of  heat  need  not  be  discussed  here  ;  it  is  a 
large  subject  by  itself. 


*  This  is  another  example  of  the  extraordinary  looseness 
of  terminology  that  runs  through  thermodynamics.  They 
ought  to  be  colds  of  combustion  or  combination,  for  the  heat 
of  fusion  or  vaporisation  is  the  heat  taken  in  during  the 
change.  The  so-called  heat  of  chemical  change  is  given  out 
during  the  change. 


136  APPENDIX. 

Heat  Conduction. — If  a  body  is  conducting  heat 
steadily,  for  instance,  a  uniform  bar  with  one  end 
kept  hot  and  the  other  cold  and  no  side  leakage,  the 
bar  can  be  cut  up  into  lengths  by  cross  cuts  imagined 
so  that  each  cut  is  at  uniform  temperature.  Let 
one  be  at  #1  and  the  next  at  02>  and  let  heat  H  pass 
per  second.  Every  second  H/0j.  comes  in  and  H/02 
comes  out,  so  that  there  is  a  volume  growth  of 
entropy  depending  on  the  conductivity  and  the 
temperature  gradient.  Any  body  whatever  which  is 
not  at  uniform  temperature  can  be  divided  up  into 
elements  bounded  by  two  sides  each  with  no  tempera- 
ture gradient,  and  a  bounding  surface  parallel  to 
the  flow  of  heat,  so  that  there  is  no  passage  of  heat 
across.  If  the  flux  of  heat  is  H  joules  per  second, 

and  0  +  —  d6  and  &  the  temperatures  of  the  two 
faces,  the  increase  of  uncompensated  entropy  per 

TT       7/j 

second,  is  —   —  y-  per  unit  volume,   and   the   un- 
6  as 

gained  work  is  —  Hdti/ds  per  second,  or  the  ungained 
power  is  -  ~H.d6/ds  watts  ;  dBjds  being  negative. 
Integration  throughout  the  volume  gives  the  total 
ungained  power. 

Units. — The  fact  that  the  units  in  thermodynamics 
have  no  names  goes  to  show  that  the  science  is  not 
fully  developed.  Measurement  is  an  essential  of 
science. 

There  is  no  name  for  the  unit  of  difference  of 


UNITS.  137 

temperature.  Trigonometry  and  heat  alone  use 
"  degrees."  Marks  or  notches  would  not  be  less 
barbarous.  Then  there  is  no  name  for  differences 
of  temperature  according  to  the  absolute  Kelvin 
scale. 

The  practical  unit  of  heat,  on  the  C.G.S.  system, 
is  the  joule,  but  this  is  hardly  ever  used  in  thermo- 
dynamics. The  calorie  is  a  survival  of  the  days 
when  it  was  not  fully  realised  that  heat  is  energy. 
It  involves  an  unnecessary  and  troublesome  co-effi- 
cient, and  people  are  putting  big  calories  and  little 
calories,  and,  perhaps,  intermediate  calories  into 
circulation. 

There  is  no  unit  of  entropy.  I  would  suggest  the 
claus ;  a  claus  being  the  entropy  which  incurs  a 
waste  of  1  joule  at  a  lowest  available  temperature  of 
unity  ;  e.g.,  if  the  lowest  available  temperature  is 
200  absolute,  and  the  entropy  10,  the  incurred 
waste  is  2,000  joules. 

Conclusion. — Enough  has  now  been  said  to  show 
the  importance  of  making  an  inherently  difficult 
subject  as  easy  to  understand  as  possible  ;  and  it  is 
hoped  that  the  somewhat  novel  way  of  arranging 
and  treating  the  subject-matter  of  the  groundwork 
of  thermodynamics  may  meet  with  the  approval  of 
those  who  specially  deal  with  that  science. 


INDEX. 


ADIABATIC  Curve 
Board  of  Trade  Unit 
British  Thermal  Unit 
Carnot        

„       Cycle 


PAGE 
...  54 
...  58 
...  58 
...  114 
...  14 
104, 137 

Clausing      25,61,114 

Compensated  Entropy  ...  9 
Degradation  of  Energy  ...  6 
Dissipation  of  Energy  ...  6 
Duhem  ...  70,  131,  133 

Energy,  Free         133 

Entrop        104 

Entropy,  Definition     8,  34,  125 
„         Factor     of     In- 
curred Waste...     22 
„         Localisation  of...     28 
„         Movement  of    ...     94 
not  a  Factor  of 

Heat 7 

Specific ...         ...     29 

Equilibrium  ...          ...   130 

Factors  of  Heat     ...         67,134 
First     Law     of     Thermo- 
dynamics ...         ...     13 

Free  Energy          133 

Gibbs  ...      61,71,122,133 

Heat,  Definition 124 

„      Diagram       ...  8,  73 

„      Factors  of    ...        67,134 

Heat- Weight         2 

Helmholtz 71,  122 

Horstmann ...   122 

Incurred  Waste  ...  22,  52 
Irreversible  Expansion  ...  42 
Isentropic 55 


PAGE 

Kelvin        53 

van  Laar    ...         ...  131 

Latent  Heat          ...        67,  115 
Localisation  of  Entropy  ...     28 

Lost  Work 85 

Massieu       7^  122 

Maxwell     ...         ...         ...7  61 

Peabody      61 

Perpetual  Motion...        11.  124 
Planck        ...         ...  131 

Rankine      35,  61 

Ilayleigh     122 

Reeve          6i 

Reversibility          26 

Sankey        59 

Second-class  Perpetual  Mo- 
tion             12 

Second  Law    of    Thermo- 
dynamics         ...     13 

Sensible  Heat        H5 

Specific  Entropy 29 

„       Heat         ...          58,  67 

Stabilities '133 

Thermodynamic   Function    35 
„  Potential...  71, 131 

Thermodynamics,  Laws  of     13 
Third    Law    of     Thermo- 
dynamics ...         ...     13 

Turbine       91 

Uncompensaled  Work     70,  131 
Ungained  Work     ...          70,85 

Waste  7,52,125 

Watt  and  State  Diagrams     49 

Work  6 

„      Lost 85 

,,      Ungained    ...         ...     70 


/ 


OF  THE 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 
LOAN  DEPT. 


This  book  is  due  on  the  last  date  stamped  below,  or 
on  the  date  to  which  renewed. 
Renewed  books  are  subject  to  immediate  recall. 

O 

nni  26  1978 

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LC.  cm.   MAR  i  6  1973 

rnoiA    «;n^  o  '*a                                     General  Library 
^l^foHTe'B  58                                Unh«,gy«f01ifcrnl. 

VB  09763 


vs 


-7? 


